{"id":15310,"date":"2019-09-05T12:07:00","date_gmt":"2019-09-05T16:07:00","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/distance-and-midpoint-formulas-circles-2\/"},"modified":"2019-09-05T12:07:00","modified_gmt":"2019-09-05T16:07:00","slug":"distance-and-midpoint-formulas-circles-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/distance-and-midpoint-formulas-circles-2\/","title":{"raw":"Distance and Midpoint Formulas; Circles","rendered":"Distance and Midpoint Formulas; Circles"},"content":{"raw":"[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Use the Distance Formula<\/li><li>Use the Midpoint Formula<\/li><li>Write the equation of a circle in standard form<\/li><li>Graph a circle<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1169148200557\" class=\"be-prepared\"><p id=\"fs-id1169145525871\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1169143576101\" type=\"1\"><li>Find the length of the hypotenuse of a right triangle whose legs are 12 and 16 inches.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4#fs-id1167832054640\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Factor: \\({x}^{2}-18x+81.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358#fs-id1167835345249\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve by completing the square: \\({x}^{2}-12x-12=0.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/f045a37e-bf7c-4818-95a1-e29172da48b4#fs-id1167836717133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><p id=\"fs-id1169147816562\">In this chapter we will be looking at the conic sections, usually called the conics, and their properties. The conics are curves that result from a plane intersecting a double cone\u2014two cones placed point-to-point. Each half of a double cone is called a nappe.<\/p><span data-type=\"media\" id=\"fs-id1169148185329\" data-alt=\"This figure shows two cones placed point to point. They are labeled nappes.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two cones placed point to point. They are labeled nappes.\" \/><\/span><p id=\"fs-id1169147842107\">There are four conics\u2014the <span data-type=\"term\">circle<\/span>, <span data-type=\"term\">parabola<\/span>, <span data-type=\"term\">ellipse<\/span>, and <span data-type=\"term\">hyperbola<\/span>. The next figure shows how the plane intersecting the double cone results in each curve.<\/p><span data-type=\"media\" id=\"fs-id1169147855585\" data-alt=\"Each of these four figures shows a double cone intersected by a plane. In the first figure, the plane is perpendicular to the axis of the cones and intersects the bottom cone to form a circle. In the second figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it intersects the base as well. Thus, the curve formed by the intersection is open at both ends. This is labeled parabola. In the third figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it does not intersect the base of the cone. Thus, the curve formed by the intersection is a closed loop, labeled ellipse. In the fourth figure, the plane is parallel to the axis, intersecting both cones. This is labeled hyperbola.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Each of these four figures shows a double cone intersected by a plane. In the first figure, the plane is perpendicular to the axis of the cones and intersects the bottom cone to form a circle. In the second figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it intersects the base as well. Thus, the curve formed by the intersection is open at both ends. This is labeled parabola. In the third figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it does not intersect the base of the cone. Thus, the curve formed by the intersection is a closed loop, labeled ellipse. In the fourth figure, the plane is parallel to the axis, intersecting both cones. This is labeled hyperbola.\" \/><\/span><p id=\"fs-id1169148059884\">Each of the curves has many applications that affect your daily life, from your cell phone to acoustics and navigation systems. In this section we will look at the properties of a circle.<\/p><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169145970589\"><h3 data-type=\"title\">Use the Distance Formula<\/h3><p id=\"fs-id1169145507316\">We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra\u2014which opens up a world of opportunities for application.<\/p><p id=\"fs-id1169143579500\">Our first step is to develop a formula to find distances between points on the rectangular coordinate system. We will plot the points and create a right triangle much as we did when we found slope in <a href=\"\/contents\/4d690921-1182-4ad3-86e0-7f849efbd233\" class=\"target-chapter\">Graphs and Functions<\/a>. We then take it one step further and use the Pythagorean Theorem to find the length of the hypotenuse of the triangle\u2014which is the distance between the points.<\/p><div data-type=\"example\" id=\"fs-id1169145493144\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147716157\"><div data-type=\"problem\" id=\"fs-id1169143581458\"><p id=\"fs-id1169148237153\">Use the rectangular coordinate system to find the distance between the points \\(\\left(6,4\\right)\\) and \\(\\left(2,1\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147777899\"><table id=\"fs-id1169148200270\" class=\"unnumbered unstyled can-break\" summary=\"Plot the two points 2, 1 and 6, 4. Connect the two points with a line d. Draw a right triangle as if you were going to find slope of that line. Find the length of each leg. The rise is 3 and the length is 4. Use the Pythagorean Theorem to find d, the distance between the two points. 3 squared plus 4 squared is d squared. Simplifying, we get 25 equals d squared. Use the Square Root Property, d is 5 or minus 5. Since distance, d is positive, we can eliminate minus 5. The distance between the points 6, 4 and 2, 1 is 5.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Plot the two points. Connect the two points<span data-type=\"newline\"><br \/><\/span>with a line.<span data-type=\"newline\"><br \/><\/span> Draw a right triangle as if you were going to<span data-type=\"newline\"><br \/><\/span>find slope.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146012839\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the length of each leg.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148229923\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Pythagorean Theorem to find <em data-effect=\"italics\">d<\/em>, the<span data-type=\"newline\"><br \/><\/span>distance between the two points.<\/td><td data-valign=\"top\" data-align=\"center\">\\({a}^{2}+{b}^{2}={c}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td><td data-valign=\"top\" data-align=\"center\">\\({3}^{2}+{4}^{2}={d}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.3em}{0ex}}9+16={d}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{2em}{0ex}}25={d}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1em}{0ex}}d=5\\phantom{\\rule{2em}{0ex}}\\overline{)d=-5}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since distance, <em data-effect=\"italics\">d<\/em> is positive, we can eliminate<span data-type=\"newline\"><br \/><\/span>\\(d=-5.\\)<\/td><td data-valign=\"top\" data-align=\"left\">The distance between the points \\(\\left(6,4\\right)\\) and<span data-type=\"newline\"><br \/><\/span>\\(\\left(2,1\\right)\\) is 5.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145575143\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169143305917\"><div data-type=\"problem\" id=\"fs-id1169147736126\"><p id=\"fs-id1169147744813\">Use the rectangular coordinate system to find the distance between the points \\(\\left(6,1\\right)\\) and \\(\\left(2,-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148224762\"><p id=\"fs-id1169147738791\">\\(d=5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145606508\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147867018\"><div data-type=\"problem\" id=\"fs-id1169147906326\"><p id=\"fs-id1169145747745\">Use the rectangular coordinate system to find the distance between the points \\(\\left(5,3\\right)\\) and \\(\\left(-3,-3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169142297030\"><p id=\"fs-id1169145844237\">\\(d=10\\)<\/p><\/div><\/div><\/div><span data-type=\"media\" id=\"fs-id1169147731429\" data-alt=\"Figure shows a graph with a right triangle. The hypotenuse connects two points, (2, 1) and (6, 4). These are respectively labeled (x1, y1) and (x2, y2). The rise is y2 minus y1, which is 4 minus 1 equals 3. The run is x2 minus x1, which is 6 minus 2 equals 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a graph with a right triangle. The hypotenuse connects two points, (2, 1) and (6, 4). These are respectively labeled (x1, y1) and (x2, y2). The rise is y2 minus y1, which is 4 minus 1 equals 3. The run is x2 minus x1, which is 6 minus 2 equals 4.\" \/><\/span><p id=\"fs-id1169147861248\">The method we used in the last example leads us to the formula to find the distance between the two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right).\\)<\/p><p id=\"fs-id1169142417464\">When we found the length of the horizontal leg we subtracted \\(6-2\\) which is \\({x}_{2}-{x}_{1}.\\)<\/p><p id=\"fs-id1169147808000\">When we found the length of the vertical leg we subtracted \\(4-1\\) which is \\({y}_{2}-{y}_{1}.\\)<\/p><p id=\"fs-id1169145721702\">If the triangle had been in a different position, we may have subtracted \\({x}_{1}-{x}_{2}\\) or \\({y}_{1}-{y}_{2}.\\) The expressions \\({x}_{2}-{x}_{1}\\) and \\({x}_{1}-{x}_{2}\\) vary only in the sign of the resulting number. To get the positive value-since distance is positive- we can use absolute value. So to generalize we will say \\(|{x}_{2}-{x}_{1}|\\) and \\(|{y}_{2}-{y}_{1}|.\\)<\/p><p id=\"fs-id1169148114458\">In the Pythagorean Theorem, we substitute the general expressions \\(|{x}_{2}-{x}_{1}|\\) and \\(|{y}_{2}-{y}_{1}|\\) rather than the numbers.<\/p><p id=\"fs-id1169143579720\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {a}^{2}&amp; +\\hfill &amp; {b}^{2}={c}^{2}\\hfill \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\hfill {\\left(|{x}_{2}-{x}_{1}|\\right)}^{2}&amp; +\\hfill &amp; {\\left(|{y}_{2}-{y}_{1}|\\right)}^{2}={d}^{2}\\hfill \\\\ \\begin{array}{c}\\text{Squaring the expressions makes them}\\hfill \\\\ \\text{positive, so we eliminate the absolute value}\\hfill \\\\ \\text{bars.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill {\\left({x}_{2}-{x}_{1}\\right)}^{2}&amp; +\\hfill &amp; {\\left({y}_{2}-{y}_{1}\\right)}^{2}={d}^{2}\\hfill \\\\ \\text{Use the Square Root Property.}\\hfill &amp; &amp; &amp; \\hfill d&amp; =\\hfill &amp; \u00b1\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ \\begin{array}{c}\\text{Distance is positive, so eliminate the negative}\\hfill \\\\ \\text{value.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill d&amp; =\\hfill &amp; \\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1169147707484\">This is the Distance Formula we use to find the distance <em data-effect=\"italics\">d<\/em> between the two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right).\\)<\/p><div data-type=\"note\" id=\"fs-id1169148205827\"><div data-type=\"title\">Distance Formula<\/div><p id=\"fs-id1169145619990\">The distance <em data-effect=\"italics\">d<\/em> between the two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right)\\) is<\/p><div data-type=\"equation\" id=\"fs-id1169147875872\" class=\"unnumbered\" data-label=\"\">\\(d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1169145759684\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145520291\"><div data-type=\"problem\" id=\"fs-id1169148234513\"><p id=\"fs-id1169148198413\">Use the Distance Formula to find the distance between the points \\(\\left(-5,-3\\right)\\) and \\(\\left(7,2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147827536\"><p id=\"fs-id1169143662536\">\\(\\begin{array}{cccccccc}\\text{Write the Distance Formula.}\\hfill &amp; &amp; &amp; &amp; &amp; d\\hfill &amp; =\\hfill &amp; \\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ \\text{Label the points,}\\phantom{\\rule{0.2em}{0ex}}\\left(\\stackrel{{x}_{1},{y}_{1}}{-5,-3}\\right),\\left(\\stackrel{{x}_{2},{y}_{2}}{7,2}\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and substitute.}\\hfill &amp; &amp; &amp; &amp; &amp; d\\hfill &amp; =\\hfill &amp; \\sqrt{{\\left(7-\\left(-5\\right)\\right)}^{2}+{\\left(2-\\left(-3\\right)\\right)}^{2}}\\hfill \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; d\\hfill &amp; =\\hfill &amp; \\sqrt{{12}^{2}+{5}^{2}}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; d\\hfill &amp; =\\hfill &amp; \\sqrt{144+25}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; d\\hfill &amp; =\\hfill &amp; \\sqrt{169}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; d\\hfill &amp; =\\hfill &amp; 13\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145640392\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147866518\"><div data-type=\"problem\" id=\"fs-id1169147741886\"><p id=\"fs-id1169147865168\">Use the Distance Formula to find the distance between the points \\(\\left(-4,-5\\right)\\) and \\(\\left(5,7\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145607126\"><p id=\"fs-id1169142139024\">\\(d=15\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148116600\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147740077\"><div data-type=\"problem\" id=\"fs-id1169147905423\"><p id=\"fs-id1169147910306\">Use the Distance Formula to find the distance between the points \\(\\left(-2,-5\\right)\\) and \\(\\left(-14,-10\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147709970\"><p id=\"fs-id1169147986537\">\\(d=13\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1169147835469\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147745025\"><div data-type=\"problem\" id=\"fs-id1169147966820\"><p id=\"fs-id1169145736696\">Use the Distance Formula to find the distance between the points \\(\\left(10,-4\\right)\\) and \\(\\left(-1,5\\right).\\) Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145673020\"><p id=\"fs-id1169147861926\">\\(\\begin{array}{cccccc}\\text{Write the Distance Formula.}\\hfill &amp; &amp; &amp; &amp; &amp; d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ \\\\ \\\\ \\text{Label the points,}\\phantom{\\rule{0.2em}{0ex}}\\left(\\stackrel{{x}_{1},{y}_{1}}{10,-4}\\right),\\left(\\stackrel{{x}_{2},{y}_{2}}{-1,5}\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and substitute.}\\hfill &amp; &amp; &amp; &amp; &amp; d=\\sqrt{{\\left(-1-10\\right)}^{2}+{\\left(5-\\left(-4\\right)\\right)}^{2}}\\hfill \\\\ \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; d=\\sqrt{{\\left(-11\\right)}^{2}+{9}^{2}}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; d=\\sqrt{121+81}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; d=\\sqrt{202}\\hfill \\\\ \\begin{array}{c}\\text{Since 202 is not a perfect square, we can leave}\\hfill \\\\ \\text{the answer in exact form or find a decimal}\\hfill \\\\ \\text{approximation.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}d=\\sqrt{202}\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}\\text{or}\\hfill \\\\ d\\approx 14.2\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147804549\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169141522177\"><div data-type=\"problem\" id=\"fs-id1169147850654\"><p id=\"fs-id1169147820803\">Use the Distance Formula to find the distance between the points \\(\\left(-4,-5\\right)\\) and \\(\\left(3,4\\right).\\) Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143580607\"><p id=\"fs-id1169142418173\">\\(d=\\sqrt{130},d\\approx 11.4\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147750284\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145672269\"><div data-type=\"problem\" id=\"fs-id1169148199563\"><p id=\"fs-id1169147710918\">Use the Distance Formula to find the distance between the points \\(\\left(-2,-5\\right)\\) and \\(\\left(-3,-4\\right).\\) Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145733064\"><p id=\"fs-id1169147831329\">\\(d=\\sqrt{2},d\\approx 1.4\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169145607214\"><h3 data-type=\"title\">Use the Midpoint Formula<\/h3><p id=\"fs-id1169148054583\">It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the <em data-effect=\"italics\">x<\/em>-coordinates and the average of the <em data-effect=\"italics\">y<\/em>-coordinates of the endpoints.<\/p><div data-type=\"note\" id=\"fs-id1169147910624\"><div data-type=\"title\">Midpoint Formula<\/div><p id=\"fs-id1169145578907\">The midpoint of the line segment whose endpoints are the two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right)\\) is<\/p><div data-type=\"equation\" id=\"fs-id1169145682904\" class=\"unnumbered\" data-label=\"\">\\(\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\)<\/div><p id=\"fs-id1169145760029\">To find the midpoint of a line segment, we find the average of the <em data-effect=\"italics\">x<\/em>-coordinates and the average of the <em data-effect=\"italics\">y<\/em>-coordinates of the endpoints.<\/p><\/div><div data-type=\"example\" id=\"fs-id1169147741999\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147964168\"><div data-type=\"problem\" id=\"fs-id1169145735350\"><p id=\"fs-id1169148132847\">Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are \\(\\left(-5,-4\\right)\\) and \\(\\left(7,2\\right).\\) Plot the endpoints and the midpoint on a rectangular coordinate system.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147962947\"><table id=\"fs-id1169147706573\" class=\"unnumbered unstyled can-break\" summary=\"Write the Midpoint Formula. The x coordinate is open parentheses x subscript 1 plus x subscript 2 close parentheses upon 2 and the y coordinate is open parentheses y subscript 1 plus y subscript 2 close parentheses upon 2. Label the points: (negative 5, negative 4) is (x subscript 1, y subscript 1) and (7, 2) is (x subscript 2, y subscript 2). Substituting these values in the formula and simplifying, we get (1, negative 1). This is the midpoint of the segment. Plot the endpoints and midpoint.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the Midpoint Formula.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Label the points, \\(\\left(\\stackrel{{x}_{1},{y}_{1}}{-5,-4}\\right),\\left(\\stackrel{{x}_{2},{y}_{2}}{7,2}\\right)\\)<span data-type=\"newline\"><br \/><\/span>and substitute.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(\\frac{-5+7}{2},\\frac{-4+2}{2}\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(\\frac{2}{2},\\frac{-2}{2}\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(1,-1\\right)\\)<span data-type=\"newline\"><br \/><\/span>The midpoint of the segment is the point<span data-type=\"newline\"><br \/><\/span>\\(\\left(1,-1\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Plot the endpoints and midpoint.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147808232\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145573908\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145545027\"><div data-type=\"problem\" id=\"fs-id1169147868541\"><p id=\"fs-id1169147873376\">Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are \\(\\left(-3,-5\\right)\\) and \\(\\left(5,7\\right).\\) Plot the endpoints and the midpoint on a rectangular coordinate system.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147816772\"><span data-type=\"media\" id=\"fs-id1169147806145\" data-alt=\"This graph shows a line segment with endpoints (negative 3, negative 5) and (5, 7) and midpoint (1, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a line segment with endpoints (negative 3, negative 5) and (5, 7) and midpoint (1, negative 1).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147949370\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147946079\"><div data-type=\"problem\" id=\"fs-id1169147719363\"><p id=\"fs-id1169147715352\">Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are \\(\\left(-2,-5\\right)\\) and \\(\\left(6,-1\\right).\\) Plot the endpoints and the midpoint on a rectangular coordinate system.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148054409\"><span data-type=\"media\" id=\"fs-id1169147832930\" data-alt=\"This graph shows a line segment with endpoints (negative 2, negative 5) and (6, negative 1) and midpoint (2, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a line segment with endpoints (negative 2, negative 5) and (6, negative 1) and midpoint (2, negative 3).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1169147852268\">Both the Distance Formula and the Midpoint Formula depend on two points, \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right).\\) It is easy to confuse which formula requires addition and which subtraction of the coordinates. If we remember where the formulas come from, is may be easier to remember the formulas.<\/p><span data-type=\"media\" id=\"fs-id1169148098752\" data-alt=\"The distance formula is d equals square root of open parentheses x2 minus x1 close parentheses squared plus open parentheses y2 minus y1 close parentheses squared end of root. This is labeled subtract the coordinates. The midpoint formula is open parentheses open parentheses x1 plus x2 close parentheses upon 2 comma open parentheses y1 plus y2 close parentheses upon 2 close parentheses. This is labeled add the coordinates.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The distance formula is d equals square root of open parentheses x2 minus x1 close parentheses squared plus open parentheses y2 minus y1 close parentheses squared end of root. This is labeled subtract the coordinates. The midpoint formula is open parentheses open parentheses x1 plus x2 close parentheses upon 2 comma open parentheses y1 plus y2 close parentheses upon 2 close parentheses. This is labeled add the coordinates.\" \/><\/span><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147770758\"><h3 data-type=\"title\">Write the Equation of a Circle in Standard Form<\/h3><p id=\"fs-id1169147744283\">As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.<\/p><span data-type=\"media\" id=\"fs-id1169147743751\" data-alt=\"This figure shows a double cone and an intersecting plane, which form a circle.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_007_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a double cone and an intersecting plane, which form a circle.\" \/><\/span><p id=\"fs-id1169147768104\">We define a <span data-type=\"term\" class=\"no-emphasis\">circle<\/span> as all points in a plane that are a fixed distance from a given point in the plane. The given point is called the <em data-effect=\"italics\">center,<\/em> \\(\\left(h,k\\right),\\) and the fixed distance is called the <em data-effect=\"italics\">radius<\/em>, <em data-effect=\"italics\">r<\/em>, of the circle.<\/p><div data-type=\"note\" id=\"fs-id1169147824880\"><div data-type=\"title\">Circle<\/div><p id=\"fs-id1169148081096\">A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the <strong data-effect=\"bold\">center<\/strong>, \\(\\left(h,k\\right),\\) and the fixed distance is called the <strong data-effect=\"bold\">radius<\/strong>, <em data-effect=\"italics\">r<\/em>, of the circle.<\/p><\/div><table id=\"fs-id1169145604491\" class=\"unnumbered unstyled\" summary=\"We look at a circle in the rectangular coordinate system. The radius is the distance from the center, (h, k) to a point on the circle, x, y. To derive the equation of a circle, we can use the distance formula with the point (h, k), point x, y and the distance, r. The distance formula is d equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared. Substituting the values and squaring both sides, we get r squared equals open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared. This is the standard form of the equation of a circle with center, (h, k) and radius, r.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We look at a circle in the rectangular coordinate system.<span data-type=\"newline\"><br \/><\/span>The radius is the distance from the center, \\(\\left(h,k\\right),\\) to a<span data-type=\"newline\"><br \/><\/span>point on the circle, \\(\\left(x,y\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147836169\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To derive the equation of a circle, we can use the<span data-type=\"newline\"><br \/><\/span>distance formula with the points \\(\\left(h,k\\right),\\) \\(\\left(x,y\\right)\\) and the<span data-type=\"newline\"><br \/><\/span>distance, <em data-effect=\"italics\">r<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.55em}{0ex}}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the values.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.55em}{0ex}}r=\\sqrt{{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Square both sides.<\/td><td data-valign=\"top\" data-align=\"left\">\\({r}^{2}={\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169147949966\">This is the standard form of the equation of a circle with center, \\(\\left(h,k\\right),\\) and radius, <em data-effect=\"italics\">r<\/em>.<\/p><div data-type=\"note\" id=\"fs-id1169148226694\"><div data-type=\"title\">Standard Form of the Equation a Circle<\/div><p id=\"fs-id1169147965627\">The standard form of the equation of a circle with center, \\(\\left(h,k\\right),\\) and radius, <em data-effect=\"italics\">r<\/em>, is<\/p><span data-type=\"media\" id=\"fs-id1169148236417\" data-alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\" \/><\/span><\/div><div data-type=\"example\" id=\"fs-id1169147850691\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147960917\"><div data-type=\"problem\" id=\"fs-id1169147878438\"><p id=\"fs-id1169141471969\">Write the standard form of the equation of the circle with radius 3 and center \\(\\left(0,0\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143519404\"><table id=\"fs-id1169145578312\" class=\"unnumbered unstyled\" summary=\"Use the standard form of the equation of a circle open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared. Substitute values r equals 3, h equals 0 and k equals 0 and simplify. We get x squared plus y squared equals 9.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a circle<\/td><td data-valign=\"top\" data-align=\"center\">\\({\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}={r}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the values \\(r=3,h=0,\\) and \\(k=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\({\\left(x-0\\right)}^{2}+{\\left(y-0\\right)}^{2}={3}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147966141\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\({x}^{2}+{y}^{2}=9\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169141221624\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169143581314\"><div data-type=\"problem\" id=\"fs-id1169147735234\"><p id=\"fs-id1169147977838\">Write the standard form of the equation of the circle with a radius of 6 and center \\(\\left(0,0\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148107155\"><p id=\"fs-id1169145661491\">\\({x}^{2}+{y}^{2}=36\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148048744\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147776946\"><div data-type=\"problem\" id=\"fs-id1169145670517\"><p id=\"fs-id1169147988048\">Write the standard form of the equation of the circle with a radius of 8 and center \\(\\left(0,0\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147910234\"><p id=\"fs-id1169141408493\">\\({x}^{2}+{y}^{2}=64\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169148096774\">In the last example, the center was \\(\\left(0,0\\right).\\) Notice what happened to the equation. Whenever the center is \\(\\left(0,0\\right),\\) the standard form becomes \\({x}^{2}+{y}^{2}={r}^{2}.\\)<\/p><div data-type=\"example\" id=\"fs-id1169142400561\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147783920\"><div data-type=\"problem\" id=\"fs-id1169147796580\"><p id=\"fs-id1169145659535\">Write the standard form of the equation of the circle with radius 2 and center \\(\\left(-1,3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147860258\"><table id=\"fs-id1169148106355\" class=\"unnumbered unstyled\" summary=\"Use the standard form of the equation of a circle open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared. Substitute values r equals 2, h equals minus 1 and k equals 3 and simplify. We get open parentheses x plus 1 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a<span data-type=\"newline\"><br \/><\/span>circle.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1.3em}{0ex}}{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}={r}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td><td data-valign=\"top\" data-align=\"center\">\\({\\left(x-\\left(-1\\right)\\right)}^{2}+{\\left(y-3\\right)}^{2}={2}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143614174\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.9em}{0ex}}{\\left(x+1\\right)}^{2}+{\\left(y-3\\right)}^{2}=4\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145547944\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145641242\"><div data-type=\"problem\" id=\"fs-id1169148125779\"><p id=\"fs-id1169142122853\">Write the standard form of the equation of the circle with a radius of 7 and center \\(\\left(2,-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141036780\"><p id=\"fs-id1169143550446\">\\({\\left(x-2\\right)}^{2}+{\\left(y+4\\right)}^{2}=49\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147960170\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147830913\"><div data-type=\"problem\" id=\"fs-id1169145670704\"><p id=\"fs-id1169147855102\">Write the standard form of the equation of the circle with a radius of 9 and center \\(\\left(-3,-5\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145733485\"><p id=\"fs-id1169148123884\">\\({\\left(x+3\\right)}^{2}+{\\left(y+5\\right)}^{2}=81\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926756510\">In the next example, the radius is not given. To calculate the radius, we use the Distance Formula with the two given points.<\/p><div data-type=\"example\" id=\"fs-id1169148225258\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148129294\"><div data-type=\"problem\" id=\"fs-id1169145600806\"><p id=\"fs-id1169147862928\">Write the standard form of the equation of the circle with center \\(\\left(2,4\\right)\\) that also contains the point \\(\\left(-2,1\\right).\\)<\/p><span data-type=\"media\" id=\"fs-id1169147846364\" data-alt=\"This graph shows circle with center at (2, 4, radius 5 and a point on the circle minus 2, 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (2, 4, radius 5 and a point on the circle minus 2, 1.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1169147860192\"><p id=\"fs-id1169143505614\">The radius is the distance from the center to any point on the circle so we can use the distance formula to calculate it. We will use the center \\(\\left(2,4\\right)\\) and point \\(\\left(-2,1\\right)\\)<\/p><p id=\"fs-id1169143459020\">\\(\\begin{array}{cccc}\\text{Use the Distance Formula to find the radius.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}r=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ \\text{Substitute the values.}\\phantom{\\rule{0.2em}{0ex}}\\left(\\stackrel{{x}_{1},{y}_{1}}{2,4}\\right),\\left(\\stackrel{{x}_{2},{y}_{2}}{-2,1}\\right)\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}r=\\sqrt{{\\left(-2-2\\right)}^{2}+{\\left(1-4\\right)}^{2}}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}r=\\sqrt{{\\left(-4\\right)}^{2}+{\\left(-3\\right)}^{2}}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}r=\\sqrt{16+9}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}r=\\sqrt{25}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}r=5\\hfill \\end{array}\\)<\/p><p id=\"fs-id1169147766768\">Now that we know the radius, \\(r=5,\\) and the center, \\(\\left(2,4\\right),\\) we can use the standard form of the equation of a circle to find the equation.<\/p><p id=\"fs-id1169145622285\">\\(\\begin{array}{cccccc}\\text{Use the standard form of the equation of a circle.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2em}{0ex}}\\hfill {\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}&amp; =\\hfill &amp; {r}^{2}\\hfill \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2em}{0ex}}{\\left(x-2\\right)}^{2}+{\\left(y-4\\right)}^{2}\\hfill &amp; =\\hfill &amp; {5}^{2}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2em}{0ex}}{\\left(x-2\\right)}^{2}+{\\left(y-4\\right)}^{2}\\hfill &amp; =\\hfill &amp; 25\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147961696\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169141545535\"><div data-type=\"problem\" id=\"fs-id1169141545537\"><p id=\"fs-id1169148231312\">Write the standard form of the equation of the circle with center \\(\\left(2,1\\right)\\) that also contains the point \\(\\left(-2,-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147828007\"><p id=\"fs-id1169147828010\">\\({\\left(x-2\\right)}^{2}+{\\left(y-1\\right)}^{2}=25\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145685239\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147857828\"><div data-type=\"problem\" id=\"fs-id1169147865098\"><p id=\"fs-id1169148116139\">Write the standard form of the equation of the circle with center \\(\\left(7,1\\right)\\) that also contains the point \\(\\left(-1,-5\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147802857\"><p id=\"fs-id1169145714523\">\\({\\left(x-7\\right)}^{2}+{\\left(y-1\\right)}^{2}=100\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169145661367\"><h3 data-type=\"title\">Graph a Circle<\/h3><p id=\"fs-id1169145779995\">Any equation of the form \\({\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}={r}^{2}\\) is the standard form of the equation of a <span data-type=\"term\" class=\"no-emphasis\">circle<\/span> with center, \\(\\left(h,k\\right),\\) and radius, <em data-effect=\"italics\">r.<\/em> We can then graph the circle on a rectangular coordinate system.<\/p><p id=\"fs-id1169145497610\">Note that the standard form calls for subtraction from <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. In the next example, the equation has \\(x+2,\\) so we need to rewrite the addition as subtraction of a negative.<\/p><div data-type=\"example\" id=\"fs-id1169147796682\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147796684\"><div data-type=\"problem\" id=\"fs-id1169147859261\"><p id=\"fs-id1169145536462\">Find the center and radius, then graph the circle: \\({\\left(x+2\\right)}^{2}+{\\left(y-1\\right)}^{2}=9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143728478\"><table id=\"fs-id1169147767591\" class=\"unnumbered unstyled\" summary=\"The equation is open parentheses x plus 2 close parentheses squared plus open parentheses y minus 1 close parentheses squared equals 9. Using the standard form of the equation of a circle open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared, we identify h equals minus 2, k equals 1 and r equals 3. Graph the circle with center at (negative 2, 1) and a radius of 3.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147836568\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a circle.<span data-type=\"newline\"><br \/><\/span>Identify the center, \\(\\left(h,k\\right)\\) and radius, <em data-effect=\"italics\">r<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147910798\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">Center: \\(\\left(-2,1\\right)\\) radius: 3<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147966350\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147982714\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147981600\"><div data-type=\"problem\" id=\"fs-id1169147725527\"><p id=\"fs-id1169141408690\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({\\left(x-3\\right)}^{2}+{\\left(y+4\\right)}^{2}=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148252442\"><p id=\"fs-id1169147962214\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(3,-4\\right)\\) with a radius of 2.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169145664917\" data-alt=\"This graph shows a circle with center at (3, negative 4) and a radius of 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a circle with center at (3, negative 4) and a radius of 2.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147906249\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147774851\"><div data-type=\"problem\" id=\"fs-id1169147745679\"><p id=\"fs-id1169147745681\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({\\left(x-3\\right)}^{2}+{\\left(y-1\\right)}^{2}=16.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147835324\"><p id=\"fs-id1169147837255\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(3,1\\right)\\) with a radius of 4.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169141172340\" data-alt=\"This graph shows circle with center at (3, 1) and a radius of 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (3, 1) and a radius of 4.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1169147850924\">To find the center and radius, we must write the equation in standard form. In the next example, we must first get the coefficient of \\({x}^{2},{y}^{2}\\) to be one.<\/p><div data-type=\"example\" id=\"fs-id1169147700684\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147709341\"><div data-type=\"problem\" id=\"fs-id1169147709344\"><p id=\"fs-id1169148232309\">Find the center and radius and then graph the circle, \\(4{x}^{2}+4{y}^{2}=64.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147867713\"><table id=\"fs-id1169143576710\" class=\"unnumbered unstyled can-break\" summary=\"The equation is 4 x squared plus 4 y squared equals 64. Dividing each side by 4, we get x squared plus y squared equals 16. Using the standard form of equation for a circle, we identify center (0, 0) and a radius of 4. Finally, we graph the circle.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147843223\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide each side by 4.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169142357711\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a circle.<span data-type=\"newline\"><br \/><\/span>Identify the center, \\(\\left(h,k\\right)\\) and radius, <em data-effect=\"italics\">r<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145499822\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">Center: \\(\\left(0,0\\right)\\) radius: 4<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169143482144\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169141171400\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147731417\"><div data-type=\"problem\" id=\"fs-id1169147758129\"><p id=\"fs-id1169147758131\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\(3{x}^{2}+3{y}^{2}=27\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141522355\"><p id=\"fs-id1169147722169\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(0,0\\right)\\) with a radius of 3.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147837044\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 3.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145578206\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145578210\"><div data-type=\"problem\" id=\"fs-id1169147868659\"><p id=\"fs-id1169147868661\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\(5{x}^{2}+5{y}^{2}=125\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147825280\"><p id=\"fs-id1169147825282\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(0,0\\right)\\) with a radius of 5.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169143662101\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 5.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1169141172490\">If we expand the equation from <a href=\"#fs-id1169147796682\" class=\"autogenerated-content\">(Figure)<\/a>, \\({\\left(x+2\\right)}^{2}+{\\left(y-1\\right)}^{2}=9,\\) the equation of the circle looks very different.<\/p><p id=\"fs-id1169143613606\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {\\left(x+2\\right)}^{2}+{\\left(y-1\\right)}^{2}&amp; =\\hfill &amp; 9\\hfill \\\\ \\text{Square the binomials.}\\hfill &amp; &amp; &amp; \\hfill {x}^{2}+4x+4+{y}^{2}-2y+1&amp; =\\hfill &amp; 9\\hfill \\\\ \\begin{array}{c}\\text{Arrange the terms in descending degree order,}\\hfill \\\\ \\text{and get zero on the right}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill {x}^{2}+{y}^{2}+4x-2y-4&amp; =\\hfill &amp; 0\\hfill \\end{array}\\)<\/p><p id=\"fs-id1169147720045\">This form of the equation is called the general form of the equation of the <span data-type=\"term\" class=\"no-emphasis\">circle<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1169145597004\"><div data-type=\"title\">General Form of the Equation of a Circle<\/div><p id=\"fs-id1169143534113\">The general form of the equation of a circle is<\/p><div data-type=\"equation\" id=\"fs-id1169147855326\" class=\"unnumbered\" data-label=\"\">\\({x}^{2}+{y}^{2}+ax+by+c=0\\)<\/div><\/div><p id=\"fs-id1169147860469\">If we are given an equation in general form, we can change it to standard form by completing the squares in both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. Then we can graph the circle using its center and radius.<\/p><div data-type=\"example\" id=\"fs-id1169148232419\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148232422\"><div data-type=\"problem\" id=\"fs-id1169145982855\"><p id=\"fs-id1169145982857\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({x}^{2}+{y}^{2}-4x-6y+4=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148063181\"><p id=\"fs-id1169148063183\">We need to rewrite this general form into standard form in order to find the center and radius.<span data-type=\"newline\"><br \/><\/span> <\/p><table id=\"fs-id1169148099878\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x squared plus y squared minus 4 x minus 6 y plus 4 equals 0. Group the x terms and y terms and collect the constants on the right side. Complete the squares by adding 4 and 9 on both sides. The equation becomes x squared minus 4 x plus 4 plus y squared minus 6y plus 9 equals minus 4 plus 4 plus 9. Rewrite as binomial squares open parentheses x minus 2 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 9. Identify the center (2, 3) and a radius of 3. Graph the circle.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147860413\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Group the <em data-effect=\"italics\">x<\/em>-terms and <em data-effect=\"italics\">y<\/em>-terms.<span data-type=\"newline\"><br \/><\/span>Collect the constants on the right side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148207026\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Complete the squares.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147832652\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite as binomial squares.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169141238753\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the center and radius.<\/td><td data-valign=\"top\" data-align=\"center\">Center: \\(\\left(2,3\\right)\\) radius: 3<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148099528\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145716835\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169141031135\"><div data-type=\"problem\" id=\"fs-id1169141031137\"><p id=\"fs-id1169148060252\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({x}^{2}+{y}^{2}-6x-8y+9=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148071420\"><p id=\"fs-id1169148071422\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(\\text{3},\\text{4}\\right)\\) with a radius of 4.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147777831\" data-alt=\"This graph shows circle with center at (3, 4) and a radius of 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (3, 4) and a radius of 4.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147935412\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148115168\"><div data-type=\"problem\" id=\"fs-id1169148115170\"><p id=\"fs-id1169143763476\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({x}^{2}+{y}^{2}+6x-2y+1=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143442555\"><p id=\"fs-id1169147963304\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(\\text{\u2212}\\text{3},\\text{1}\\right)\\) with a radius of 3.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147849082\" data-alt=\"This graph shows circle with center at (negative 3, 1) and a radius of 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 3, 1) and a radius of 3.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1169145545022\">In the next example, there is a <em data-effect=\"italics\">y<\/em>-term and a \\({y}^{2}\\)-term. But notice that there is no <em data-effect=\"italics\">x<\/em>-term, only an \\({x}^{2}\\)-term. We have seen this before and know that it means <em data-effect=\"italics\">h<\/em> is 0. We will need to complete the square for the <em data-effect=\"italics\">y<\/em> terms, but not for the <em data-effect=\"italics\">x<\/em> terms.<\/p><div data-type=\"example\" id=\"fs-id1169145669033\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145669035\"><div data-type=\"problem\" id=\"fs-id1169145608091\"><p id=\"fs-id1169145608093\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({x}^{2}+{y}^{2}+8y=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148231288\"><p id=\"fs-id1169148231290\">We need to rewrite this general form into standard form in order to find the center and radius.<span data-type=\"newline\"><br \/><\/span> <\/p><table id=\"fs-id1169147750696\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x squared plus y squared plus 8 y equals 0. Group the x terms and y terms. There are no constants to collect on the right side. Add 16 on both sides to complete the square term y squared plus 8 y. Rewrite as binomial squares. The center is (0, negative 4) and radius is 4. Graph the circle.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148211519\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Group the <em data-effect=\"italics\">x<\/em>-terms and <em data-effect=\"italics\">y<\/em>-terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145645275\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">There are no constants to collect on the<span data-type=\"newline\"><br \/><\/span>right side.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Complete the square for \\({y}^{2}+8y.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169143763644\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite as binomial squares.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147746020\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the center and radius.<\/td><td data-valign=\"top\" data-align=\"center\">Center: \\(\\left(0,-4\\right)\\) radius: 4<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147836113\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145730864\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147712428\"><div data-type=\"problem\" id=\"fs-id1169147712430\"><p id=\"fs-id1169145672392\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({x}^{2}+{y}^{2}-2x-3=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147721994\"><p id=\"fs-id1169147721996\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(-1,0\\right)\\) with a radius of 2.<span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147873387\" data-alt=\"This graph shows circle with center at (1, 0) and a radius of 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (1, 0) and a radius of 2.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145496484\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147810598\"><div data-type=\"problem\" id=\"fs-id1169147810601\"><p id=\"fs-id1169147709373\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: \\({x}^{2}+{y}^{2}-12y+11=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145574492\"><p id=\"fs-id1169145574494\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(0,6\\right)\\) with a radius of 5.<span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169145777347\" data-alt=\"This graph shows circle with center at (0, 6) and a radius of 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 6) and a radius of 5.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145977482\" class=\"media-2\"><p id=\"fs-id1169145579605\">Access these online resources for additional instructions and practice with using the distance and midpoint formulas, and graphing circles.<\/p><ul id=\"fs-id1169141299165\" data-bullet-style=\"bullet\"><li><a href=\"https:\/\/openstax.org\/l\/37distmidcircle\">Distance-Midpoint Formulas and Circles<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37distmid2pts\">Finding the Distance and Midpoint Between Two Points<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37stformcircle\">Completing the Square to Write Equation in Standard Form of a Circle<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169147960019\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1169147737903\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Distance Formula:<\/strong> The distance <em data-effect=\"italics\">d<\/em> between the two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right)\\) is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1169145661301\" class=\"unnumbered\" data-label=\"\">\\(d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\)<\/div><\/li><li><strong data-effect=\"bold\">Midpoint Formula:<\/strong> The midpoint of the line segment whose endpoints are the two points \\(\\left({x}_{1},{y}_{1}\\right)\\) and \\(\\left({x}_{2},{y}_{2}\\right)\\) is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1169148100172\" class=\"unnumbered\" data-label=\"\">\\(\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\)<\/div><span data-type=\"newline\"><br \/><\/span> To find the midpoint of a line segment, we find the average of the <em data-effect=\"italics\">x<\/em>-coordinates and the average of the <em data-effect=\"italics\">y<\/em>-coordinates of the endpoints.<\/li><li><strong data-effect=\"bold\">Circle:<\/strong> A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the <em data-effect=\"italics\">center,<\/em> \\(\\left(h,k\\right),\\) and the fixed distance is called the <em data-effect=\"italics\">radius, r,<\/em> of the circle.<\/li><li><strong data-effect=\"bold\">Standard Form of the Equation a Circle:<\/strong> The standard form of the equation of a circle with center, \\(\\left(h,k\\right),\\) and radius, <em data-effect=\"italics\">r,<\/em> is<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1169145639520\" data-alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\" \/><\/span><\/li><li><strong data-effect=\"bold\">General Form of the Equation of a Circle:<\/strong> The general form of the equation of a circle is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1169148233474\" class=\"unnumbered\" data-label=\"\">\\({x}^{2}+{y}^{2}+ax+by+c=0\\)<\/div><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169145601807\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169145506799\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1169148208240\"><strong data-effect=\"bold\">Use the Distance Formula<\/strong><\/p><p id=\"fs-id1169143506078\">In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p><div data-type=\"exercise\" id=\"fs-id1169147845456\"><div data-type=\"problem\" id=\"fs-id1169147845458\"><p id=\"fs-id1169141298655\">\\(\\left(2,0\\right)\\) and \\(\\left(5,4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145536183\"><p id=\"fs-id1169145536185\">\\(d=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145778721\"><div data-type=\"problem\" id=\"fs-id1169147809343\"><p id=\"fs-id1169147809345\">\\(\\left(-4,-3\\right)\\) and \\(\\left(2,5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147739242\"><div data-type=\"problem\" id=\"fs-id1169147739244\"><p id=\"fs-id1169148081777\">\\(\\left(-4,-3\\right)\\) and \\(\\left(8,2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143660939\"><p id=\"fs-id1169143660941\">13<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143659178\"><div data-type=\"problem\" id=\"fs-id1169147855200\"><p id=\"fs-id1169147855202\">\\(\\left(-7,-3\\right)\\) and \\(\\left(8,5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145760210\"><div data-type=\"problem\" id=\"fs-id1169141500543\"><p id=\"fs-id1169141500545\">\\(\\left(-1,4\\right)\\) and \\(\\left(2,0\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147982282\"><p id=\"fs-id1169147982284\">5<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147960782\"><div data-type=\"problem\" id=\"fs-id1169147960784\"><p id=\"fs-id1169147960524\">\\(\\left(-1,3\\right)\\) and \\(\\left(5,-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145614088\"><div data-type=\"problem\" id=\"fs-id1169145614090\"><p id=\"fs-id1169145614092\">\\(\\left(1,-4\\right)\\) and \\(\\left(6,8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143317903\"><p id=\"fs-id1169143317905\">13<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147855655\"><div data-type=\"problem\" id=\"fs-id1169145674218\"><p id=\"fs-id1169145674220\">\\(\\left(-8,-2\\right)\\) and \\(\\left(7,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147962933\"><div data-type=\"problem\" id=\"fs-id1169147962936\"><p id=\"fs-id1169147988290\">\\(\\left(-3,-5\\right)\\) and \\(\\left(0,1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148231252\"><p id=\"fs-id1169148231254\">\\(76.\\phantom{\\rule{0.2em}{0ex}}d=3\\sqrt{5},d\\approx 6.7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145526198\"><div data-type=\"problem\" id=\"fs-id1169145526200\"><p id=\"fs-id1169145735103\">\\(\\left(-1,-2\\right)\\) and \\(\\left(-3,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145493751\"><div data-type=\"problem\" id=\"fs-id1169148249795\"><p id=\"fs-id1169148249797\">\\(\\left(3,-1\\right)\\) and \\(\\left(1,7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147797011\"><p id=\"fs-id1169148207801\">\\(d=\\sqrt{68},d\\approx 8.2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143637830\"><div data-type=\"problem\" id=\"fs-id1169147873390\"><p id=\"fs-id1169147873392\">\\(\\left(-4,-5\\right)\\) and \\(\\left(7,4\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169147981703\"><strong data-effect=\"bold\">Use the Midpoint Formula<\/strong><\/p><p id=\"fs-id1169147825927\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the midpoint of the line segments whose endpoints are given and <span class=\"token\">\u24d1<\/span> plot the endpoints and the midpoint on a rectangular coordinate system.<\/p><div data-type=\"exercise\" id=\"fs-id1169147935246\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147935248\"><p id=\"fs-id1169145577358\">\\(\\left(0,-5\\right)\\) and \\(\\left(4,-3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147849887\"><p id=\"fs-id1169147849889\"><span class=\"token\">\u24d0<\/span> Midpoint: \\(\\left(2,-4\\right)\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169143574297\" data-alt=\"This graph shows line segment with endpoints (0, negative 5) and (4, negative 3) and midpoint (2, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows line segment with endpoints (0, negative 5) and (4, negative 3) and midpoint (2, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147700121\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147700123\"><p id=\"fs-id1169147700125\">\\(\\left(-2,-6\\right)\\) and \\(\\left(6,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147825975\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147840099\"><p id=\"fs-id1169147840101\">\\(\\left(3,-1\\right)\\) and \\(\\left(4,-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147707833\"><p id=\"fs-id1169147776056\"><span class=\"token\">\u24d0<\/span> Midpoint: \\(\\left(3\\frac{1}{2},-1\\frac{1}{2}\\right)\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147983031\" data-alt=\"This graph shows line segment with endpoints (3, negative 1) and (4, negative 2) and midpoint (3 and a half, negative 1 and a half).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_313_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows line segment with endpoints (3, negative 1) and (4, negative 2) and midpoint (3 and a half, negative 1 and a half).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143659157\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169143659159\"><p id=\"fs-id1169143659161\">\\(\\left(-3,-3\\right)\\) and \\(\\left(6,-1\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169148200511\"><strong data-effect=\"bold\">Write the Equation of a Circle in Standard Form<\/strong><\/p><p id=\"fs-id1169147837518\">In the following exercises, write the standard form of the equation of the circle with the given radius and center \\(\\left(0,0\\right).\\)<\/p><div data-type=\"exercise\" id=\"fs-id1169147806757\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169148107175\"><p id=\"fs-id1169148107177\">Radius: 7<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147878807\"><p id=\"fs-id1169147878810\">\\({x}^{2}+{y}^{2}=49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169142479956\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169142479958\"><p id=\"fs-id1169142479961\">Radius: 9<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143686441\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169143686443\"><p id=\"fs-id1169145734173\">Radius: \\(\\sqrt{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145520570\"><p id=\"fs-id1169145520572\">\\({x}^{2}+{y}^{2}=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141036387\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169141036389\"><p id=\"fs-id1169141264477\">Radius: \\(\\sqrt{5}\\)<\/p><\/div><\/div><p id=\"fs-id1169147857763\">In the following exercises, write the standard form of the equation of the circle with the given radius and center<\/p><div data-type=\"exercise\" id=\"fs-id1169147851663\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147851665\"><p id=\"fs-id1169143580024\">Radius: 1, center: \\(\\left(3,5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147759733\"><p id=\"fs-id1169147759735\">\\({\\left(x-3\\right)}^{2}+{\\left(y-5\\right)}^{2}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143380550\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147734671\"><p id=\"fs-id1169147734673\">Radius: 10, center: \\(\\left(-2,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145716238\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145716241\"><p id=\"fs-id1169145716243\">Radius: \\(2.5,\\) center: \\(\\left(1.5,-3.5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147876835\"><p id=\"fs-id1169147819644\">\\({\\left(x-1.5\\right)}^{2}+{\\left(y+3.5\\right)}^{2}=6.25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148185356\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169148185358\"><p id=\"fs-id1169147833498\">Radius: \\(1.5,\\) center: \\(\\left(-5.5,-6.5\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169148132049\">For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.<\/p><div data-type=\"exercise\" id=\"fs-id1169148132053\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169148200547\"><p id=\"fs-id1169148200549\">Center \\(\\left(3,-2\\right)\\) with point \\(\\left(3,6\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147768219\"><p id=\"fs-id1169143662778\">\\({\\left(x-3\\right)}^{2}+{\\left(y+2\\right)}^{2}=64\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147862548\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147862550\"><p id=\"fs-id1169147858336\">Center \\(\\left(6,-6\\right)\\) with point \\(\\left(2,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147742617\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147742619\"><p id=\"fs-id1169147866448\">Center \\(\\left(4,4\\right)\\) with point \\(\\left(2,2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147878186\"><p id=\"fs-id1169147878188\">\\({\\left(x-4\\right)}^{2}+{\\left(y-4\\right)}^{2}=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145544916\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145544918\"><p id=\"fs-id1169145544921\">Center \\(\\left(-5,6\\right)\\) with point \\(\\left(-2,3\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169148037424\"><strong data-effect=\"bold\">Graph a Circle<\/strong><\/p><p id=\"fs-id1169148037429\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the center and radius, then <span class=\"token\">\u24d1<\/span> graph each circle.<\/p><div data-type=\"exercise\" id=\"fs-id1169145780272\"><div data-type=\"problem\" id=\"fs-id1169148084941\"><p id=\"fs-id1169148084944\">\\({\\left(x+5\\right)}^{2}+{\\left(y+3\\right)}^{2}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145493485\"><p id=\"fs-id1169145493487\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(-5,-3\\right)\\) with a radius of 1.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169145728541\" data-alt=\"This graph shows a circle with center at (negative 5, negative 3) and a radius of 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_315_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a circle with center at (negative 5, negative 3) and a radius of 1.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147963087\"><div data-type=\"problem\" id=\"fs-id1169147963089\"><p id=\"fs-id1169147963091\">\\({\\left(x-2\\right)}^{2}+{\\left(y-3\\right)}^{2}=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143580870\"><div data-type=\"problem\" id=\"fs-id1169143580873\"><p id=\"fs-id1169143580875\">\\({\\left(x-4\\right)}^{2}+{\\left(y+2\\right)}^{2}=16\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145620168\"><p id=\"fs-id1169143459000\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(4,-2\\right)\\) with a radius of 4.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169145663017\" data-alt=\"This graph shows circle with center at (4, negative 2) and a radius of 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_317_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (4, negative 2) and a radius of 4.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147824357\"><div data-type=\"problem\" id=\"fs-id1169147824359\"><p id=\"fs-id1169148103813\">\\({\\left(x+2\\right)}^{2}+{\\left(y-5\\right)}^{2}=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143581746\"><div data-type=\"problem\" id=\"fs-id1169147879538\"><p id=\"fs-id1169147879540\">\\({x}^{2}+{\\left(y+2\\right)}^{2}=25\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147767026\"><p id=\"fs-id1169145668600\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(0,-2\\right)\\) with a radius of 5.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169148125063\" data-alt=\"This graph shows circle with center at (negative 2, 5) and a radius of 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_319_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 2, 5) and a radius of 5.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147848673\"><div data-type=\"problem\" id=\"fs-id1169147848675\"><p id=\"fs-id1169147848677\">\\({\\left(x-1\\right)}^{2}+{y}^{2}=36\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145720376\"><div data-type=\"problem\" id=\"fs-id1169143575137\"><p id=\"fs-id1169143575139\">\\({\\left(x-1.5\\right)}^{2}+{\\left(y+2.5\\right)}^{2}=0.25\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145777437\"><p id=\"fs-id1169145777439\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(1.5,2.5\\right)\\) with a radius of \\(0.5.\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147725203\" data-alt=\"This graph shows circle with center at (1.5, 2.5) and a radius of 0.5\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_321_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (1.5, 2.5) and a radius of 0.5\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145579182\"><div data-type=\"problem\" id=\"fs-id1169145579184\"><p id=\"fs-id1169145579187\">\\({\\left(x-1\\right)}^{2}+{\\left(y-3\\right)}^{2}=\\frac{9}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145731596\"><div data-type=\"problem\" id=\"fs-id1169145731598\"><p id=\"fs-id1169147741233\">\\({x}^{2}+{y}^{2}=64\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147707766\"><p id=\"fs-id1169147707768\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(0,0\\right)\\) with a radius of 8.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147770825\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 8.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_323_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 8.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145578236\"><div data-type=\"problem\" id=\"fs-id1169145578238\"><p id=\"fs-id1169145578240\">\\({x}^{2}+{y}^{2}=49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145577421\"><div data-type=\"problem\" id=\"fs-id1169145660340\"><p id=\"fs-id1169145660342\">\\(2{x}^{2}+2{y}^{2}=8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145733520\"><p id=\"fs-id1169145733522\"><span class=\"token\">\u24d0<\/span> The circle is centered at \\(\\left(0,0\\right)\\) with a radius of 2.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169148133834\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_325_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 2.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147854265\"><div data-type=\"problem\" id=\"fs-id1169147854268\"><p id=\"fs-id1169147854270\">\\(6{x}^{2}+6{y}^{2}=216\\)<\/p><\/div><\/div><p id=\"fs-id1169147965199\">In the following exercises, <span class=\"token\">\u24d0<\/span> identify the center and radius and <span class=\"token\">\u24d1<\/span> graph.<\/p><div data-type=\"exercise\" id=\"fs-id1169143575851\"><div data-type=\"problem\" id=\"fs-id1169143575853\"><p id=\"fs-id1169148081242\">\\({x}^{2}+{y}^{2}+2x+6y+9=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147967370\"><p id=\"fs-id1169147967372\"><span class=\"token\">\u24d0<\/span> Center: \\(\\left(-1,-3\\right),\\) radius: 1<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147875802\" data-alt=\"This graph shows circle with center at (negative 1, negative 3) and a radius of 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_327_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 1, negative 3) and a radius of 1.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143579426\"><div data-type=\"problem\" id=\"fs-id1169143579428\"><p id=\"fs-id1169143579430\">\\({x}^{2}+{y}^{2}-6x-8y=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147864093\"><div data-type=\"problem\" id=\"fs-id1169147864095\"><p id=\"fs-id1169147864098\">\\({x}^{2}+{y}^{2}-4x+10y-7=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145730625\"><p id=\"fs-id1169145730628\"><span class=\"token\">\u24d0<\/span> Center: \\(\\left(2,-5\\right),\\) radius: 6<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147758639\" data-alt=\"This graph shows circle with center at (2, negative 5) and a radius of 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_329_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (2, negative 5) and a radius of 6.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148063172\"><div data-type=\"problem\" id=\"fs-id1169147804002\"><p id=\"fs-id1169147804005\">\\({x}^{2}+{y}^{2}+12x-14y+21=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145663824\"><div data-type=\"problem\" id=\"fs-id1169145663826\"><p id=\"fs-id1169147856895\">\\({x}^{2}+{y}^{2}+6y+5=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147960756\"><p id=\"fs-id1169147960758\"><span class=\"token\">\u24d0<\/span> Center: \\(\\left(0,-3\\right),\\) radius: 2<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147879986\" data-alt=\"This graph shows circle with center at (0, negative 3) and a radius of 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_331_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, negative 3) and a radius of 2.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147805025\"><div data-type=\"problem\" id=\"fs-id1169147741808\"><p id=\"fs-id1169147741810\">\\({x}^{2}+{y}^{2}-10y=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148231447\"><div data-type=\"problem\" id=\"fs-id1169143580427\"><p id=\"fs-id1169143580429\">\\({x}^{2}+{y}^{2}+4x=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147856831\"><p id=\"fs-id1169147856833\"><span class=\"token\">\u24d0<\/span> Center: \\(\\left(-2,0\\right),\\) radius: = 2 <span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1169147794371\" data-alt=\"This graph shows circle with center at (negative 2, 0) and a radius of 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_333_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 2, 0) and a radius of 2.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145493979\"><div data-type=\"problem\" id=\"fs-id1169145605845\"><p id=\"fs-id1169145605847\">\\({x}^{2}+{y}^{2}-14x+13=0\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1169147873338\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1169147764974\"><div data-type=\"problem\" id=\"fs-id1169147864549\"><p id=\"fs-id1169147864551\">Explain the relationship between the distance formula and the equation of a circle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143575555\"><p id=\"fs-id1169143575557\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147878929\"><div data-type=\"problem\" id=\"fs-id1169147878931\"><p id=\"fs-id1169145731327\">Is a circle a function? Explain why or why not.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148208053\"><div data-type=\"problem\" id=\"fs-id1169147965394\"><p id=\"fs-id1169147965396\">In your own words, state the definition of a circle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143349074\"><p id=\"fs-id1169143349076\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148072593\"><div data-type=\"problem\" id=\"fs-id1169148072595\"><p id=\"fs-id1169147830310\">In your own words, explain the steps you would take to change the general form of the equation of a circle to the standard form.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169145499937\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1169147851532\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1169147834300\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><p id=\"fs-id1169147708321\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p><p id=\"fs-id1169145644689\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p><p id=\"fs-id1169147978241\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p><p id=\"fs-id1169143575150\">\u2026no - I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1169145731792\"><dt>circle<\/dt><dd id=\"fs-id1169147879662\">A circle is all points in a plane that are a fixed distance from a fixed point in the plane.<\/dd><\/dl><\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Use the Distance Formula<\/li>\n<li>Use the Midpoint Formula<\/li>\n<li>Write the equation of a circle in standard form<\/li>\n<li>Graph a circle<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148200557\" class=\"be-prepared\">\n<p id=\"fs-id1169145525871\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1169143576101\" type=\"1\">\n<li>Find the length of the hypotenuse of a right triangle whose legs are 12 and 16 inches.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4#fs-id1167832054640\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d995e1ec564c58b47d1692020aa11d95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#43;&#56;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358#fs-id1167835345249\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-685814ec72331ee3e4f75e7b5aff88dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#49;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"143\" style=\"vertical-align: -1px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/f045a37e-bf7c-4818-95a1-e29172da48b4#fs-id1167836717133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1169147816562\">In this chapter we will be looking at the conic sections, usually called the conics, and their properties. The conics are curves that result from a plane intersecting a double cone\u2014two cones placed point-to-point. Each half of a double cone is called a nappe.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148185329\" data-alt=\"This figure shows two cones placed point to point. They are labeled nappes.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two cones placed point to point. They are labeled nappes.\" \/><\/span><\/p>\n<p id=\"fs-id1169147842107\">There are four conics\u2014the <span data-type=\"term\">circle<\/span>, <span data-type=\"term\">parabola<\/span>, <span data-type=\"term\">ellipse<\/span>, and <span data-type=\"term\">hyperbola<\/span>. The next figure shows how the plane intersecting the double cone results in each curve.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147855585\" data-alt=\"Each of these four figures shows a double cone intersected by a plane. In the first figure, the plane is perpendicular to the axis of the cones and intersects the bottom cone to form a circle. In the second figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it intersects the base as well. Thus, the curve formed by the intersection is open at both ends. This is labeled parabola. In the third figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it does not intersect the base of the cone. Thus, the curve formed by the intersection is a closed loop, labeled ellipse. In the fourth figure, the plane is parallel to the axis, intersecting both cones. This is labeled hyperbola.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Each of these four figures shows a double cone intersected by a plane. In the first figure, the plane is perpendicular to the axis of the cones and intersects the bottom cone to form a circle. In the second figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it intersects the base as well. Thus, the curve formed by the intersection is open at both ends. This is labeled parabola. In the third figure, the plane is at an angle to the axis and intersects the bottom cone in such a way that it does not intersect the base of the cone. Thus, the curve formed by the intersection is a closed loop, labeled ellipse. In the fourth figure, the plane is parallel to the axis, intersecting both cones. This is labeled hyperbola.\" \/><\/span><\/p>\n<p id=\"fs-id1169148059884\">Each of the curves has many applications that affect your daily life, from your cell phone to acoustics and navigation systems. In this section we will look at the properties of a circle.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169145970589\">\n<h3 data-type=\"title\">Use the Distance Formula<\/h3>\n<p id=\"fs-id1169145507316\">We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra\u2014which opens up a world of opportunities for application.<\/p>\n<p id=\"fs-id1169143579500\">Our first step is to develop a formula to find distances between points on the rectangular coordinate system. We will plot the points and create a right triangle much as we did when we found slope in <a href=\"\/contents\/4d690921-1182-4ad3-86e0-7f849efbd233\" class=\"target-chapter\">Graphs and Functions<\/a>. We then take it one step further and use the Pythagorean Theorem to find the length of the hypotenuse of the triangle\u2014which is the distance between the points.<\/p>\n<div data-type=\"example\" id=\"fs-id1169145493144\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147716157\">\n<div data-type=\"problem\" id=\"fs-id1169143581458\">\n<p id=\"fs-id1169148237153\">Use the rectangular coordinate system to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86825c0bacfa777b8fe368820523b723_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfc029320f22ea34a0de2989d0a2e86f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147777899\">\n<table id=\"fs-id1169148200270\" class=\"unnumbered unstyled can-break\" summary=\"Plot the two points 2, 1 and 6, 4. Connect the two points with a line d. Draw a right triangle as if you were going to find slope of that line. Find the length of each leg. The rise is 3 and the length is 4. Use the Pythagorean Theorem to find d, the distance between the two points. 3 squared plus 4 squared is d squared. Simplifying, we get 25 equals d squared. Use the Square Root Property, d is 5 or minus 5. Since distance, d is positive, we can eliminate minus 5. The distance between the points 6, 4 and 2, 1 is 5.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Plot the two points. Connect the two points<span data-type=\"newline\"><br \/><\/span>with a line.<span data-type=\"newline\"><br \/><\/span> Draw a right triangle as if you were going to<span data-type=\"newline\"><br \/><\/span>find slope.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146012839\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the length of each leg.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148229923\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Pythagorean Theorem to find <em data-effect=\"italics\">d<\/em>, the<span data-type=\"newline\"><br \/><\/span>distance between the two points.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ea02600b9972a334e42686c1b0b4cf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fbe98786b3a505ca9f7c32fde946d3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5543d2e0dab4cd0137baa2b2721c7d59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#43;&#49;&#54;&#61;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba01da2d4ab574ffd5c86df8d7c5b454_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#61;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfdc216d2bb727f0f8be5fb427297468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#61;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#100;&#61;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since distance, <em data-effect=\"italics\">d<\/em> is positive, we can eliminate<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cd47a2de95d5a26ae68f08ada0d00da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">The distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86825c0bacfa777b8fe368820523b723_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is 5.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145575143\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169143305917\">\n<div data-type=\"problem\" id=\"fs-id1169147736126\">\n<p id=\"fs-id1169147744813\">Use the rectangular coordinate system to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc9db18ceda8b325515059e9c425b44f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-534204d07513941720df89b471c28252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148224762\">\n<p id=\"fs-id1169147738791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313a91835988d3729267cb47e609f373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145606508\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147867018\">\n<div data-type=\"problem\" id=\"fs-id1169147906326\">\n<p id=\"fs-id1169145747745\">Use the rectangular coordinate system to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-159ccfbcee3ed270fafd4d79974b76e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87a9d0dcd2df40301ef7c7ba22c76548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169142297030\">\n<p id=\"fs-id1169145844237\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9019367815b67e4e7a859391801b1e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><span data-type=\"media\" id=\"fs-id1169147731429\" data-alt=\"Figure shows a graph with a right triangle. The hypotenuse connects two points, (2, 1) and (6, 4). These are respectively labeled (x1, y1) and (x2, y2). The rise is y2 minus y1, which is 4 minus 1 equals 3. The run is x2 minus x1, which is 6 minus 2 equals 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a graph with a right triangle. The hypotenuse connects two points, (2, 1) and (6, 4). These are respectively labeled (x1, y1) and (x2, y2). The rise is y2 minus y1, which is 4 minus 1 equals 3. The run is x2 minus x1, which is 6 minus 2 equals 4.\" \/><\/span><\/p>\n<p id=\"fs-id1169147861248\">The method we used in the last example leads us to the formula to find the distance between the two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4de3b1dcfe0feb0fab20411f044405f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169142417464\">When we found the length of the horizontal leg we subtracted <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07de14c1afb064b4c70b27baec07d954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/> which is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e2847bfe3305b6b147fe9aa41b21555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169147808000\">When we found the length of the vertical leg we subtracted <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c63bbabcfdb3cbfd34f1ed0324e66b76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: -1px;\" \/> which is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-607e4fc433fcdc4144b6abe0b8b25805_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169145721702\">If the triangle had been in a different position, we may have subtracted <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-998e9d6d97a4549c9b1688abee0f52d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-948d1346594f8ddc51a92277c0b58290_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: -4px;\" \/> The expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6da0a1c7afe2bcf6c90d30e18fb4498_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-998e9d6d97a4549c9b1688abee0f52d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: -4px;\" \/> vary only in the sign of the resulting number. To get the positive value-since distance is positive- we can use absolute value. So to generalize we will say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af79fa510e980e55a9a6b165dba03d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80194c4f2edb1115c655314963f4a5e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169148114458\">In the Pythagorean Theorem, we substitute the general expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af79fa510e980e55a9a6b165dba03d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1db0d790308951dfda25ac7c025bb12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> rather than the numbers.<\/p>\n<p id=\"fs-id1169143579720\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebb15f0ed92df687cd13d27e6f2b25e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#124;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#124;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#124;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#124;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#113;&#117;&#97;&#114;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#115;&#32;&#109;&#97;&#107;&#101;&#115;&#32;&#116;&#104;&#101;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#32;&#115;&#111;&#32;&#119;&#101;&#32;&#101;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#97;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#83;&#113;&#117;&#97;&#114;&#101;&#32;&#82;&#111;&#111;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#100;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#105;&#115;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#32;&#115;&#111;&#32;&#101;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#97;&#108;&#117;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#100;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"181\" width=\"749\" style=\"vertical-align: -83px;\" \/><\/p>\n<p id=\"fs-id1169147707484\">This is the Distance Formula we use to find the distance <em data-effect=\"italics\">d<\/em> between the two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4de3b1dcfe0feb0fab20411f044405f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1169148205827\">\n<div data-type=\"title\">Distance Formula<\/div>\n<p id=\"fs-id1169145619990\">The distance <em data-effect=\"italics\">d<\/em> between the two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147875872\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e765dcb2bc6b649e1b6465aeece39a30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"228\" style=\"vertical-align: -10px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169145759684\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145520291\">\n<div data-type=\"problem\" id=\"fs-id1169148234513\">\n<p id=\"fs-id1169148198413\">Use the Distance Formula to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb242707b07d2123762ae0b5253ad3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-482e00d66c072f8a8aeee2be5e830848_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147827536\">\n<p id=\"fs-id1169143662536\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3d67a481ac77ace0dc302053997470c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#68;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#70;&#111;&#114;&#109;&#117;&#108;&#97;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#97;&#98;&#101;&#108;&#32;&#116;&#104;&#101;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#45;&#53;&#44;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#55;&#44;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#49;&#50;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#43;&#50;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"182\" width=\"762\" style=\"vertical-align: -84px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145640392\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147866518\">\n<div data-type=\"problem\" id=\"fs-id1169147741886\">\n<p id=\"fs-id1169147865168\">Use the Distance Formula to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3840972c083bab5f065585197b233498_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afcf723209959a9e4b2544d525594201_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145607126\">\n<p id=\"fs-id1169142139024\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a652579076674377bf797349bf82267b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148116600\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147740077\">\n<div data-type=\"problem\" id=\"fs-id1169147905423\">\n<p id=\"fs-id1169147910306\">Use the Distance Formula to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc65316aa33a9c85778a50c7aec6891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da65b030344aa2f388ef6b3167e16627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#44;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147709970\">\n<p id=\"fs-id1169147986537\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-309dffaf13740494ec38d903929617ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169147835469\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147745025\">\n<div data-type=\"problem\" id=\"fs-id1169147966820\">\n<p id=\"fs-id1169145736696\">Use the Distance Formula to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60baf13d9a34514cd3d1a4d4f2a43a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bde486fae93e3280e17ec3032001a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145673020\">\n<p id=\"fs-id1169147861926\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ec90c5d3594186e6e76158daf971ea4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#68;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#70;&#111;&#114;&#109;&#117;&#108;&#97;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#97;&#98;&#101;&#108;&#32;&#116;&#104;&#101;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#49;&#48;&#44;&#45;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#45;&#49;&#44;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#57;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#43;&#56;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#48;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#50;&#48;&#50;&#32;&#105;&#115;&#32;&#110;&#111;&#116;&#32;&#97;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#44;&#32;&#119;&#101;&#32;&#99;&#97;&#110;&#32;&#108;&#101;&#97;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#105;&#110;&#32;&#101;&#120;&#97;&#99;&#116;&#32;&#102;&#111;&#114;&#109;&#32;&#111;&#114;&#32;&#102;&#105;&#110;&#100;&#32;&#97;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#112;&#112;&#114;&#111;&#120;&#105;&#109;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#48;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#100;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#49;&#52;&#46;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"303\" width=\"738\" style=\"vertical-align: -145px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147804549\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169141522177\">\n<div data-type=\"problem\" id=\"fs-id1169147850654\">\n<p id=\"fs-id1169147820803\">Use the Distance Formula to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3840972c083bab5f065585197b233498_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3d8af68a49dc3005e869e1b2f3518b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143580607\">\n<p id=\"fs-id1169142418173\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ac2e4287b91755ced3259b3eb41ab13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#51;&#48;&#125;&#44;&#100;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#49;&#49;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147750284\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145672269\">\n<div data-type=\"problem\" id=\"fs-id1169148199563\">\n<p id=\"fs-id1169147710918\">Use the Distance Formula to find the distance between the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc65316aa33a9c85778a50c7aec6891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34d359cbab98b576a127c13db3c8e58d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145733064\">\n<p id=\"fs-id1169147831329\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cb9c83531e80ea2eba3fc97d9210c52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#100;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#49;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169145607214\">\n<h3 data-type=\"title\">Use the Midpoint Formula<\/h3>\n<p id=\"fs-id1169148054583\">It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the <em data-effect=\"italics\">x<\/em>-coordinates and the average of the <em data-effect=\"italics\">y<\/em>-coordinates of the endpoints.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147910624\">\n<div data-type=\"title\">Midpoint Formula<\/div>\n<p id=\"fs-id1169145578907\">The midpoint of the line segment whose endpoints are the two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1169145682904\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1bfe816d29cd1ca9d500b4c4002de6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1169145760029\">To find the midpoint of a line segment, we find the average of the <em data-effect=\"italics\">x<\/em>-coordinates and the average of the <em data-effect=\"italics\">y<\/em>-coordinates of the endpoints.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169147741999\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147964168\">\n<div data-type=\"problem\" id=\"fs-id1169145735350\">\n<p id=\"fs-id1169148132847\">Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8aa5693f7456de7e35690e5d2138859_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-482e00d66c072f8a8aeee2be5e830848_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> Plot the endpoints and the midpoint on a rectangular coordinate system.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147962947\">\n<table id=\"fs-id1169147706573\" class=\"unnumbered unstyled can-break\" summary=\"Write the Midpoint Formula. The x coordinate is open parentheses x subscript 1 plus x subscript 2 close parentheses upon 2 and the y coordinate is open parentheses y subscript 1 plus y subscript 2 close parentheses upon 2. Label the points: (negative 5, negative 4) is (x subscript 1, y subscript 1) and (7, 2) is (x subscript 2, y subscript 2). Substituting these values in the formula and simplifying, we get (1, negative 1). This is the midpoint of the segment. Plot the endpoints and midpoint.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the Midpoint Formula.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1bfe816d29cd1ca9d500b4c4002de6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Label the points, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01d17c698009b3db77163201726a5110_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#45;&#53;&#44;&#45;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#55;&#44;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"142\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/><\/span>and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c16ed0b120ccc1f3f9e695da0bce36f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#53;&#43;&#55;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#43;&#50;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42b22afb7a454c31c1b6d66e01c7849f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3ac38dbb39343c28a60a287dfb114b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>The midpoint of the segment is the point<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c37f8c6862300339d3b66a637d5f296f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Plot the endpoints and midpoint.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147808232\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145573908\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145545027\">\n<div data-type=\"problem\" id=\"fs-id1169147868541\">\n<p id=\"fs-id1169147873376\">Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd0b31c672d6277d527b16652fb255d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afcf723209959a9e4b2544d525594201_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> Plot the endpoints and the midpoint on a rectangular coordinate system.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147816772\"><span data-type=\"media\" id=\"fs-id1169147806145\" data-alt=\"This graph shows a line segment with endpoints (negative 3, negative 5) and (5, 7) and midpoint (1, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a line segment with endpoints (negative 3, negative 5) and (5, 7) and midpoint (1, negative 1).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147949370\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147946079\">\n<div data-type=\"problem\" id=\"fs-id1169147719363\">\n<p id=\"fs-id1169147715352\">Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc65316aa33a9c85778a50c7aec6891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-035610bc8f728460fad8c1cb19f43b9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> Plot the endpoints and the midpoint on a rectangular coordinate system.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148054409\"><span data-type=\"media\" id=\"fs-id1169147832930\" data-alt=\"This graph shows a line segment with endpoints (negative 2, negative 5) and (6, negative 1) and midpoint (2, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a line segment with endpoints (negative 2, negative 5) and (6, negative 1) and midpoint (2, negative 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147852268\">Both the Distance Formula and the Midpoint Formula depend on two points, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4de3b1dcfe0feb0fab20411f044405f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> It is easy to confuse which formula requires addition and which subtraction of the coordinates. If we remember where the formulas come from, is may be easier to remember the formulas.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148098752\" data-alt=\"The distance formula is d equals square root of open parentheses x2 minus x1 close parentheses squared plus open parentheses y2 minus y1 close parentheses squared end of root. This is labeled subtract the coordinates. The midpoint formula is open parentheses open parentheses x1 plus x2 close parentheses upon 2 comma open parentheses y1 plus y2 close parentheses upon 2 close parentheses. This is labeled add the coordinates.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The distance formula is d equals square root of open parentheses x2 minus x1 close parentheses squared plus open parentheses y2 minus y1 close parentheses squared end of root. This is labeled subtract the coordinates. The midpoint formula is open parentheses open parentheses x1 plus x2 close parentheses upon 2 comma open parentheses y1 plus y2 close parentheses upon 2 close parentheses. This is labeled add the coordinates.\" \/><\/span><\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147770758\">\n<h3 data-type=\"title\">Write the Equation of a Circle in Standard Form<\/h3>\n<p id=\"fs-id1169147744283\">As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147743751\" data-alt=\"This figure shows a double cone and an intersecting plane, which form a circle.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_007_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a double cone and an intersecting plane, which form a circle.\" \/><\/span><\/p>\n<p id=\"fs-id1169147768104\">We define a <span data-type=\"term\" class=\"no-emphasis\">circle<\/span> as all points in a plane that are a fixed distance from a given point in the plane. The given point is called the <em data-effect=\"italics\">center,<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and the fixed distance is called the <em data-effect=\"italics\">radius<\/em>, <em data-effect=\"italics\">r<\/em>, of the circle.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147824880\">\n<div data-type=\"title\">Circle<\/div>\n<p id=\"fs-id1169148081096\">A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the <strong data-effect=\"bold\">center<\/strong>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and the fixed distance is called the <strong data-effect=\"bold\">radius<\/strong>, <em data-effect=\"italics\">r<\/em>, of the circle.<\/p>\n<\/div>\n<table id=\"fs-id1169145604491\" class=\"unnumbered unstyled\" summary=\"We look at a circle in the rectangular coordinate system. The radius is the distance from the center, (h, k) to a point on the circle, x, y. To derive the equation of a circle, we can use the distance formula with the point (h, k), point x, y and the distance, r. The distance formula is d equals square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses squared. Substituting the values and squaring both sides, we get r squared equals open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared. This is the standard form of the equation of a circle with center, (h, k) and radius, r.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We look at a circle in the rectangular coordinate system.<span data-type=\"newline\"><br \/><\/span>The radius is the distance from the center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> to a<span data-type=\"newline\"><br \/><\/span>point on the circle, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ab61ec2033ed5b69d0447eac5d6a4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147836169\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To derive the equation of a circle, we can use the<span data-type=\"newline\"><br \/><\/span>distance formula with the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> and the<span data-type=\"newline\"><br \/><\/span>distance, <em data-effect=\"italics\">r<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2de96f7ae6ce72ec2ecdc6d349484654_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"227\" style=\"vertical-align: -10px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e620a5f23382439a26403fcab13db0a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"198\" style=\"vertical-align: -10px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Square both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1216f77f7300026fe3691e9e96945f4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"187\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169147949966\">This is the standard form of the equation of a circle with center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and radius, <em data-effect=\"italics\">r<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1169148226694\">\n<div data-type=\"title\">Standard Form of the Equation a Circle<\/div>\n<p id=\"fs-id1169147965627\">The standard form of the equation of a circle with center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and radius, <em data-effect=\"italics\">r<\/em>, is<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148236417\" data-alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\" \/><\/span><\/div>\n<div data-type=\"example\" id=\"fs-id1169147850691\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147960917\">\n<div data-type=\"problem\" id=\"fs-id1169147878438\">\n<p id=\"fs-id1169141471969\">Write the standard form of the equation of the circle with radius 3 and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143519404\">\n<table id=\"fs-id1169145578312\" class=\"unnumbered unstyled\" summary=\"Use the standard form of the equation of a circle open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared. Substitute values r equals 3, h equals 0 and k equals 0 and simplify. We get x squared plus y squared equals 9.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a circle<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e7a8fe4da1eed761ff259bdbe0e8cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c9b2cf1bc99c91b2310266c75c64834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#51;&#44;&#104;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad4ecb9382b22eaa970a6589c3f57064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5552b456102951ad514ac69a6cda5691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#51;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147966141\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bee4195382ea0bf825e615ffd3b49da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169141221624\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169143581314\">\n<div data-type=\"problem\" id=\"fs-id1169147735234\">\n<p id=\"fs-id1169147977838\">Write the standard form of the equation of the circle with a radius of 6 and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148107155\">\n<p id=\"fs-id1169145661491\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c0adb4c5f478031c28a7e48667300f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148048744\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147776946\">\n<div data-type=\"problem\" id=\"fs-id1169145670517\">\n<p id=\"fs-id1169147988048\">Write the standard form of the equation of the circle with a radius of 8 and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147910234\">\n<p id=\"fs-id1169141408493\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-358bac6ea7b08a21b71ed22199d52e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148096774\">In the last example, the center was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> Notice what happened to the equation. Whenever the center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> the standard form becomes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f1a394148a166f0525629933e8a2559_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1169142400561\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147783920\">\n<div data-type=\"problem\" id=\"fs-id1169147796580\">\n<p id=\"fs-id1169145659535\">Write the standard form of the equation of the circle with radius 2 and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cd74210bd97c5d0b58124465f48adf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147860258\">\n<table id=\"fs-id1169148106355\" class=\"unnumbered unstyled\" summary=\"Use the standard form of the equation of a circle open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared. Substitute values r equals 2, h equals minus 1 and k equals 3 and simplify. We get open parentheses x plus 1 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a<span data-type=\"newline\"><br \/><\/span>circle.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-057305f6094b5bb948c46bc7b1f7df4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-812e4a46fc6450709595adbeca4d1176_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#50;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"211\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143614174\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8adc6532b6a08979d6d1a28efa722dc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145547944\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145641242\">\n<div data-type=\"problem\" id=\"fs-id1169148125779\">\n<p id=\"fs-id1169142122853\">Write the standard form of the equation of the circle with a radius of 7 and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1953a0f325fcab1337240dbfff719e7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141036780\">\n<p id=\"fs-id1169143550446\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d76b56bcf6c68cda1ec3d7f905849600_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147960170\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147830913\">\n<div data-type=\"problem\" id=\"fs-id1169145670704\">\n<p id=\"fs-id1169147855102\">Write the standard form of the equation of the circle with a radius of 9 and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37fa9dbb8454d162ef24836afb0c0bf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145733485\">\n<p id=\"fs-id1169148123884\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-874522128395ba71e52864f68c960be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926756510\">In the next example, the radius is not given. To calculate the radius, we use the Distance Formula with the two given points.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148225258\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148129294\">\n<div data-type=\"problem\" id=\"fs-id1169145600806\">\n<p id=\"fs-id1169147862928\">Write the standard form of the equation of the circle with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> that also contains the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8214cd1dae89c5047c3dc6e12087cf14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147846364\" data-alt=\"This graph shows circle with center at (2, 4, radius 5 and a point on the circle minus 2, 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (2, 4, radius 5 and a point on the circle minus 2, 1.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169147860192\">\n<p id=\"fs-id1169143505614\">The radius is the distance from the center to any point on the circle so we can use the distance formula to calculate it. We will use the center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169143459020\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b57552a802068e6b14248d1a3b3e8bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#68;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#70;&#111;&#114;&#109;&#117;&#108;&#97;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#117;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#50;&#44;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#45;&#50;&#44;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#43;&#57;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"694\" style=\"vertical-align: -76px;\" \/><\/p>\n<p id=\"fs-id1169147766768\">Now that we know the radius, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c95494ce9e4e3ed092cf994c619d7f2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> and the center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffd77866f4f656944a55bcf7992eb7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> we can use the standard form of the equation of a circle to find the equation.<\/p>\n<p id=\"fs-id1169145622285\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28eb8627003af5fe420d171b301ee3db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#97;&#32;&#99;&#105;&#114;&#99;&#108;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"679\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147961696\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169141545535\">\n<div data-type=\"problem\" id=\"fs-id1169141545537\">\n<p id=\"fs-id1169148231312\">Write the standard form of the equation of the circle with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> that also contains the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6272f10a7ad1fcb0b6b804f308cf17ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147828007\">\n<p id=\"fs-id1169147828010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99876db6cc0630a5e36135af5ad814f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145685239\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147857828\">\n<div data-type=\"problem\" id=\"fs-id1169147865098\">\n<p id=\"fs-id1169148116139\">Write the standard form of the equation of the circle with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f63e6ea0c8474555192b305e3472481d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> that also contains the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af443c2890d3eeb2456fc94734bd46cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147802857\">\n<p id=\"fs-id1169145714523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f2d31a42b52ddf20ba66aa42f233749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"195\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169145661367\">\n<h3 data-type=\"title\">Graph a Circle<\/h3>\n<p id=\"fs-id1169145779995\">Any equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e7a8fe4da1eed761ff259bdbe0e8cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#114;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/> is the standard form of the equation of a <span data-type=\"term\" class=\"no-emphasis\">circle<\/span> with center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and radius, <em data-effect=\"italics\">r.<\/em> We can then graph the circle on a rectangular coordinate system.<\/p>\n<p id=\"fs-id1169145497610\">Note that the standard form calls for subtraction from <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. In the next example, the equation has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4940abc918bbd4b24c766b2dfdb2e7a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -4px;\" \/> so we need to rewrite the addition as subtraction of a negative.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147796682\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147796684\">\n<div data-type=\"problem\" id=\"fs-id1169147859261\">\n<p id=\"fs-id1169145536462\">Find the center and radius, then graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-315d76bb66c3888e7aa1a3821f95d509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143728478\">\n<table id=\"fs-id1169147767591\" class=\"unnumbered unstyled\" summary=\"The equation is open parentheses x plus 2 close parentheses squared plus open parentheses y minus 1 close parentheses squared equals 9. Using the standard form of the equation of a circle open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared, we identify h equals minus 2, k equals 1 and r equals 3. Graph the circle with center at (negative 2, 1) and a radius of 3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147836568\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a circle.<span data-type=\"newline\"><br \/><\/span>Identify the center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> and radius, <em data-effect=\"italics\">r<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147910798\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\">Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> radius: 3<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147966350\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147982714\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147981600\">\n<div data-type=\"problem\" id=\"fs-id1169147725527\">\n<p id=\"fs-id1169141408690\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfe60c6a585d5a91a313253f0a75722f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148252442\">\n<p id=\"fs-id1169147962214\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abcb6ace4b542ff0928998579e86da44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> with a radius of 2.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169145664917\" data-alt=\"This graph shows a circle with center at (3, negative 4) and a radius of 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a circle with center at (3, negative 4) and a radius of 2.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147906249\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147774851\">\n<div data-type=\"problem\" id=\"fs-id1169147745679\">\n<p id=\"fs-id1169147745681\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02da42c8d6116eeffdf6332ce6a285ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"190\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147835324\">\n<p id=\"fs-id1169147837255\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 4.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169141172340\" data-alt=\"This graph shows circle with center at (3, 1) and a radius of 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (3, 1) and a radius of 4.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147850924\">To find the center and radius, we must write the equation in standard form. In the next example, we must first get the coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d74676ef92ec909ccbc56a1de933a235_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -4px;\" \/> to be one.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147700684\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147709341\">\n<div data-type=\"problem\" id=\"fs-id1169147709344\">\n<p id=\"fs-id1169148232309\">Find the center and radius and then graph the circle, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20e573a46ecd9e8b67f81f4e08b875b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147867713\">\n<table id=\"fs-id1169143576710\" class=\"unnumbered unstyled can-break\" summary=\"The equation is 4 x squared plus 4 y squared equals 64. Dividing each side by 4, we get x squared plus y squared equals 16. Using the standard form of equation for a circle, we identify center (0, 0) and a radius of 4. Finally, we graph the circle.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147843223\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide each side by 4.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169142357711\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the standard form of the equation of a circle.<span data-type=\"newline\"><br \/><\/span>Identify the center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> and radius, <em data-effect=\"italics\">r<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145499822\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\">Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> radius: 4<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169143482144\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169141171400\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147731417\">\n<div data-type=\"problem\" id=\"fs-id1169147758129\">\n<p id=\"fs-id1169147758131\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d530ba5aa9d5846eedffe1afd9066d11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141522355\">\n<p id=\"fs-id1169147722169\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 3.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147837044\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 3.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145578206\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145578210\">\n<div data-type=\"problem\" id=\"fs-id1169147868659\">\n<p id=\"fs-id1169147868661\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-778f0cb78b94ef025ca008af9b227f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147825280\">\n<p id=\"fs-id1169147825282\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 5.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169143662101\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 5.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169141172490\">If we expand the equation from <a href=\"#fs-id1169147796682\" class=\"autogenerated-content\">(Figure)<\/a>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-240974d34905b256ac311264e83d2191_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"181\" style=\"vertical-align: -4px;\" \/> the equation of the circle looks very different.<\/p>\n<p id=\"fs-id1169143613606\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a47baee6d4ab375f18a43d74da1d1a21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#113;&#117;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#52;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#114;&#114;&#97;&#110;&#103;&#101;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#105;&#110;&#32;&#100;&#101;&#115;&#99;&#101;&#110;&#100;&#105;&#110;&#103;&#32;&#100;&#101;&#103;&#114;&#101;&#101;&#32;&#111;&#114;&#100;&#101;&#114;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#103;&#101;&#116;&#32;&#122;&#101;&#114;&#111;&#32;&#111;&#110;&#32;&#116;&#104;&#101;&#32;&#114;&#105;&#103;&#104;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#50;&#121;&#45;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"678\" style=\"vertical-align: -37px;\" \/><\/p>\n<p id=\"fs-id1169147720045\">This form of the equation is called the general form of the equation of the <span data-type=\"term\" class=\"no-emphasis\">circle<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1169145597004\">\n<div data-type=\"title\">General Form of the Equation of a Circle<\/div>\n<p id=\"fs-id1169143534113\">The general form of the equation of a circle is<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147855326\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29c7670916408ddb61c5a1b9ef9fca3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#120;&#43;&#98;&#121;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"198\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169147860469\">If we are given an equation in general form, we can change it to standard form by completing the squares in both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. Then we can graph the circle using its center and radius.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148232419\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148232422\">\n<div data-type=\"problem\" id=\"fs-id1169145982855\">\n<p id=\"fs-id1169145982857\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42bfb71224b4425ca82270216eb6d182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#54;&#121;&#43;&#52;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148063181\">\n<p id=\"fs-id1169148063183\">We need to rewrite this general form into standard form in order to find the center and radius.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<table id=\"fs-id1169148099878\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x squared plus y squared minus 4 x minus 6 y plus 4 equals 0. Group the x terms and y terms and collect the constants on the right side. Complete the squares by adding 4 and 9 on both sides. The equation becomes x squared minus 4 x plus 4 plus y squared minus 6y plus 9 equals minus 4 plus 4 plus 9. Rewrite as binomial squares open parentheses x minus 2 close parentheses squared plus open parentheses y minus 3 close parentheses squared equals 9. Identify the center (2, 3) and a radius of 3. Graph the circle.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147860413\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Group the <em data-effect=\"italics\">x<\/em>-terms and <em data-effect=\"italics\">y<\/em>-terms.<span data-type=\"newline\"><br \/><\/span>Collect the constants on the right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148207026\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Complete the squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147832652\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite as binomial squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169141238753\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the center and radius.<\/td>\n<td data-valign=\"top\" data-align=\"center\">Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> radius: 3<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148099528\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145716835\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169141031135\">\n<div data-type=\"problem\" id=\"fs-id1169141031137\">\n<p id=\"fs-id1169148060252\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-683763651f2c0e2aa052df6767416fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#56;&#121;&#43;&#57;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148071420\">\n<p id=\"fs-id1169148071422\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4fba8067d86ad05987128763ea9935f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 4.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147777831\" data-alt=\"This graph shows circle with center at (3, 4) and a radius of 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (3, 4) and a radius of 4.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147935412\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148115168\">\n<div data-type=\"problem\" id=\"fs-id1169148115170\">\n<p id=\"fs-id1169143763476\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0785ba25ea6bfb5df54ea1ba07208d70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#50;&#121;&#43;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143442555\">\n<p id=\"fs-id1169147963304\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-003fc56d4709a6c4e33dd4bca70ca8f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 3.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147849082\" data-alt=\"This graph shows circle with center at (negative 3, 1) and a radius of 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 3, 1) and a radius of 3.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145545022\">In the next example, there is a <em data-effect=\"italics\">y<\/em>-term and a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/>-term. But notice that there is no <em data-effect=\"italics\">x<\/em>-term, only an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>-term. We have seen this before and know that it means <em data-effect=\"italics\">h<\/em> is 0. We will need to complete the square for the <em data-effect=\"italics\">y<\/em> terms, but not for the <em data-effect=\"italics\">x<\/em> terms.<\/p>\n<div data-type=\"example\" id=\"fs-id1169145669033\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145669035\">\n<div data-type=\"problem\" id=\"fs-id1169145608091\">\n<p id=\"fs-id1169145608093\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0303879be76ef5e6569228e203ebb309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148231288\">\n<p id=\"fs-id1169148231290\">We need to rewrite this general form into standard form in order to find the center and radius.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<table id=\"fs-id1169147750696\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x squared plus y squared plus 8 y equals 0. Group the x terms and y terms. There are no constants to collect on the right side. Add 16 on both sides to complete the square term y squared plus 8 y. Rewrite as binomial squares. The center is (0, negative 4) and radius is 4. Graph the circle.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148211519\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Group the <em data-effect=\"italics\">x<\/em>-terms and <em data-effect=\"italics\">y<\/em>-terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145645275\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">There are no constants to collect on the<span data-type=\"newline\"><br \/><\/span>right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Complete the square for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8faa383d595f68f3be50f5b378755782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169143763644\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite as binomial squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147746020\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the center and radius.<\/td>\n<td data-valign=\"top\" data-align=\"center\">Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f3c0bd5adc0ec0fa1707d7989e5c00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> radius: 4<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the circle.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147836113\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145730864\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147712428\">\n<div data-type=\"problem\" id=\"fs-id1169147712430\">\n<p id=\"fs-id1169145672392\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67a374dac6767f9f10f5acb35882e2d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"164\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147721994\">\n<p id=\"fs-id1169147721996\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebe7ea89f522e94da67a0a0622127bab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> with a radius of 2.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147873387\" data-alt=\"This graph shows circle with center at (1, 0) and a radius of 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (1, 0) and a radius of 2.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145496484\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147810598\">\n<div data-type=\"problem\" id=\"fs-id1169147810601\">\n<p id=\"fs-id1169147709373\"><span class=\"token\">\u24d0<\/span> Find the center and radius, then <span class=\"token\">\u24d1<\/span> graph the circle: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8bd4735e03f4526045c5db41e2794a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#43;&#49;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145574492\">\n<p id=\"fs-id1169145574494\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3be590e6084f04549fae6ac7c149a1d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 5.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169145777347\" data-alt=\"This graph shows circle with center at (0, 6) and a radius of 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 6) and a radius of 5.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145977482\" class=\"media-2\">\n<p id=\"fs-id1169145579605\">Access these online resources for additional instructions and practice with using the distance and midpoint formulas, and graphing circles.<\/p>\n<ul id=\"fs-id1169141299165\" data-bullet-style=\"bullet\">\n<li><a href=\"https:\/\/openstax.org\/l\/37distmidcircle\">Distance-Midpoint Formulas and Circles<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37distmid2pts\">Finding the Distance and Midpoint Between Two Points<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37stformcircle\">Completing the Square to Write Equation in Standard Form of a Circle<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169147960019\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169147737903\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Distance Formula:<\/strong> The distance <em data-effect=\"italics\">d<\/em> between the two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1169145661301\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e765dcb2bc6b649e1b6465aeece39a30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"228\" style=\"vertical-align: -10px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Midpoint Formula:<\/strong> The midpoint of the line segment whose endpoints are the two points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf7d3000a611d6a6b02b6093c3dfb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1169148100172\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1bfe816d29cd1ca9d500b4c4002de6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/div>\n<p><span data-type=\"newline\"><br \/><\/span> To find the midpoint of a line segment, we find the average of the <em data-effect=\"italics\">x<\/em>-coordinates and the average of the <em data-effect=\"italics\">y<\/em>-coordinates of the endpoints.<\/li>\n<li><strong data-effect=\"bold\">Circle:<\/strong> A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the <em data-effect=\"italics\">center,<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and the fixed distance is called the <em data-effect=\"italics\">radius, r,<\/em> of the circle.<\/li>\n<li><strong data-effect=\"bold\">Standard Form of the Equation a Circle:<\/strong> The standard form of the equation of a circle with center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> and radius, <em data-effect=\"italics\">r,<\/em> is<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1169145639520\" data-alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">General Form of the Equation of a Circle:<\/strong> The general form of the equation of a circle is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1169148233474\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29c7670916408ddb61c5a1b9ef9fca3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#120;&#43;&#98;&#121;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"198\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169145601807\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169145506799\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169148208240\"><strong data-effect=\"bold\">Use the Distance Formula<\/strong><\/p>\n<p id=\"fs-id1169143506078\">In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147845456\">\n<div data-type=\"problem\" id=\"fs-id1169147845458\">\n<p id=\"fs-id1169141298655\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559928bd7c8949c8342dd73437aef05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f721d5513f52f88fbefa5f38e5679ddc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145536183\">\n<p id=\"fs-id1169145536185\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313a91835988d3729267cb47e609f373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145778721\">\n<div data-type=\"problem\" id=\"fs-id1169147809343\">\n<p id=\"fs-id1169147809345\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebe785a1a1f173c2cb47dc17a15cdc63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2da1eb750fc283f55cb9396d5536b47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147739242\">\n<div data-type=\"problem\" id=\"fs-id1169147739244\">\n<p id=\"fs-id1169148081777\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebe785a1a1f173c2cb47dc17a15cdc63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9b49b201a3efa56726407933e068d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143660939\">\n<p id=\"fs-id1169143660941\">13<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143659178\">\n<div data-type=\"problem\" id=\"fs-id1169147855200\">\n<p id=\"fs-id1169147855202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5db13f20e02f8d03333a782732fed899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-648acaf5866f6160ec660bee40426678_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145760210\">\n<div data-type=\"problem\" id=\"fs-id1169141500543\">\n<p id=\"fs-id1169141500545\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b098ad691d3b1e66796376e6000e2385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559928bd7c8949c8342dd73437aef05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147982282\">\n<p id=\"fs-id1169147982284\">5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147960782\">\n<div data-type=\"problem\" id=\"fs-id1169147960784\">\n<p id=\"fs-id1169147960524\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2347704000ad2e9ae878a8427611b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38deec632295a22f878bf66affc585e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145614088\">\n<div data-type=\"problem\" id=\"fs-id1169145614090\">\n<p id=\"fs-id1169145614092\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95e282826e52860edf4d8703db75fb7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6032429e4ed847fa0727a169e71d4eaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143317903\">\n<p id=\"fs-id1169143317905\">13<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147855655\">\n<div data-type=\"problem\" id=\"fs-id1169145674218\">\n<p id=\"fs-id1169145674220\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0079e38fdd16e43ef7a238df2ab245d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147962933\">\n<div data-type=\"problem\" id=\"fs-id1169147962936\">\n<p id=\"fs-id1169147988290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd0b31c672d6277d527b16652fb255d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99d11505e9b59f8e2d3351529e3354c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148231252\">\n<p id=\"fs-id1169148231254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dea6e5704b77780f22cec00cf0c7f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#54;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#61;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#44;&#100;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#54;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145526198\">\n<div data-type=\"problem\" id=\"fs-id1169145526200\">\n<p id=\"fs-id1169145735103\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce38ace23dd1b8f15095a1aff4a73f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-862a9525ac8db19009bf877fff4597b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145493751\">\n<div data-type=\"problem\" id=\"fs-id1169148249795\">\n<p id=\"fs-id1169148249797\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ec0ae5ae28a676caf47ab43b74e8d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147797011\">\n<p id=\"fs-id1169148207801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51eef50ea6767976bfa548dddded3e03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#56;&#125;&#44;&#100;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#56;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143637830\">\n<div data-type=\"problem\" id=\"fs-id1169147873390\">\n<p id=\"fs-id1169147873392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3840972c083bab5f065585197b233498_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-383413ec6e3436c8b884f20c9879bf19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147981703\"><strong data-effect=\"bold\">Use the Midpoint Formula<\/strong><\/p>\n<p id=\"fs-id1169147825927\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the midpoint of the line segments whose endpoints are given and <span class=\"token\">\u24d1<\/span> plot the endpoints and the midpoint on a rectangular coordinate system.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147935246\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147935248\">\n<p id=\"fs-id1169145577358\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f6ab1a0ed415088c10eaaa3977a4992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a73c1812707ce5b0b7341d06a667a00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147849887\">\n<p id=\"fs-id1169147849889\"><span class=\"token\">\u24d0<\/span> Midpoint: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d5b80bc57d91ba9e2936cc72d75618b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169143574297\" data-alt=\"This graph shows line segment with endpoints (0, negative 5) and (4, negative 3) and midpoint (2, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows line segment with endpoints (0, negative 5) and (4, negative 3) and midpoint (2, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147700121\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147700123\">\n<p id=\"fs-id1169147700125\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ea573d1f83d98725156be4230a519b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e228594fa0479f071602af78e20058a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147825975\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147840099\">\n<p id=\"fs-id1169147840101\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-067f891ede38d609b0614eef084ae132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147707833\">\n<p id=\"fs-id1169147776056\"><span class=\"token\">\u24d0<\/span> Midpoint: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23dbb941eab81bdfa3e0ef50e5a22ed4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#45;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"73\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147983031\" data-alt=\"This graph shows line segment with endpoints (3, negative 1) and (4, negative 2) and midpoint (3 and a half, negative 1 and a half).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_313_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows line segment with endpoints (3, negative 1) and (4, negative 2) and midpoint (3 and a half, negative 1 and a half).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143659157\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169143659159\">\n<p id=\"fs-id1169143659161\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5bcb2f0a8e6c8a4c7021c467dd9513a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e41965a17973324ca787678c930108f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148200511\"><strong data-effect=\"bold\">Write the Equation of a Circle in Standard Form<\/strong><\/p>\n<p id=\"fs-id1169147837518\">In the following exercises, write the standard form of the equation of the circle with the given radius and center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147806757\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169148107175\">\n<p id=\"fs-id1169148107177\">Radius: 7<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147878807\">\n<p id=\"fs-id1169147878810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4846d3ba91620499c56c560ff0529ae1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169142479956\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169142479958\">\n<p id=\"fs-id1169142479961\">Radius: 9<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143686441\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169143686443\">\n<p id=\"fs-id1169145734173\">Radius: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f51c9c2ccb4ec4dcca8926b645d14c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145520570\">\n<p id=\"fs-id1169145520572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0d6dc02f474a95b7757332c9fec46a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141036387\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169141036389\">\n<p id=\"fs-id1169141264477\">Radius: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ba329d3f2f86c0fb47d848202e0ee7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147857763\">In the following exercises, write the standard form of the equation of the circle with the given radius and center<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147851663\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147851665\">\n<p id=\"fs-id1169143580024\">Radius: 1, center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c1fb7b2aa4d3e58c5af2d13706f56d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147759733\">\n<p id=\"fs-id1169147759735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b440fe52f5cb91d93e529b41fd71ea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143380550\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147734671\">\n<p id=\"fs-id1169147734673\">Radius: 10, center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b61914c33c0ffe75331208eaa502d627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145716238\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145716241\">\n<p id=\"fs-id1169145716243\">Radius: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-430a22a867bdf5ec4ead5a816d16b4ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"27\" style=\"vertical-align: -4px;\" \/> center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aa4ec8342adca4cce7cc40288614eb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#46;&#53;&#44;&#45;&#51;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147876835\">\n<p id=\"fs-id1169147819644\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b53a5a660681d5c4ff329146a53e5db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"226\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148185356\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169148185358\">\n<p id=\"fs-id1169147833498\">Radius: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3dfd55690568a79d0569830709d6c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"26\" style=\"vertical-align: -4px;\" \/> center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56bdd1a107bd1d6f795f0ef67f499ef4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#46;&#53;&#44;&#45;&#54;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148132049\">For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148132053\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169148200547\">\n<p id=\"fs-id1169148200549\">Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbaec542907415eac32615dfae0ae911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> with point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1fa2be7ed8e95fd6a934ca178fab1d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147768219\">\n<p id=\"fs-id1169143662778\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d035d7649c38de08d9048c961392d330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147862548\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147862550\">\n<p id=\"fs-id1169147858336\">Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44a7150612ce1e0a067c6c62d007bc00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> with point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1564d1d2328bb6bd9e7b30e6d573d2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147742617\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147742619\">\n<p id=\"fs-id1169147866448\">Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f071f7020da53c225631b96b8f9875e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f78f604644c2cdddbea2fc4d8ad49cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147878186\">\n<p id=\"fs-id1169147878188\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-936eaf937cdfaedcfcfe694a828e715a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145544916\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145544918\">\n<p id=\"fs-id1169145544921\">Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee47278eba3e60a9253d35f5859bb2e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> with point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fd7f677a681964debbd5fb9bbb3c944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148037424\"><strong data-effect=\"bold\">Graph a Circle<\/strong><\/p>\n<p id=\"fs-id1169148037429\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the center and radius, then <span class=\"token\">\u24d1<\/span> graph each circle.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145780272\">\n<div data-type=\"problem\" id=\"fs-id1169148084941\">\n<p id=\"fs-id1169148084944\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5b083b6ac2f75566ec982579abc2fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145493485\">\n<p id=\"fs-id1169145493487\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb242707b07d2123762ae0b5253ad3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> with a radius of 1.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169145728541\" data-alt=\"This graph shows a circle with center at (negative 5, negative 3) and a radius of 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_315_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a circle with center at (negative 5, negative 3) and a radius of 1.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147963087\">\n<div data-type=\"problem\" id=\"fs-id1169147963089\">\n<p id=\"fs-id1169147963091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3d2a9911b42757757d09aea8d8fe6b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143580870\">\n<div data-type=\"problem\" id=\"fs-id1169143580873\">\n<p id=\"fs-id1169143580875\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cbfbafcbae064d64ef1d45c650eee567_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145620168\">\n<p id=\"fs-id1169143459000\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-067f891ede38d609b0614eef084ae132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> with a radius of 4.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169145663017\" data-alt=\"This graph shows circle with center at (4, negative 2) and a radius of 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_317_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (4, negative 2) and a radius of 4.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147824357\">\n<div data-type=\"problem\" id=\"fs-id1169147824359\">\n<p id=\"fs-id1169148103813\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b5c256c22383ee489579b686a2171d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143581746\">\n<div data-type=\"problem\" id=\"fs-id1169147879538\">\n<p id=\"fs-id1169147879540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1807c6fb124a05c5164e550e122e1153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147767026\">\n<p id=\"fs-id1169145668600\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ecef9f206503704c74407265b403ee3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> with a radius of 5.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148125063\" data-alt=\"This graph shows circle with center at (negative 2, 5) and a radius of 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_319_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 2, 5) and a radius of 5.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147848673\">\n<div data-type=\"problem\" id=\"fs-id1169147848675\">\n<p id=\"fs-id1169147848677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4e5789859f959d0cb35d689c8aefdf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145720376\">\n<div data-type=\"problem\" id=\"fs-id1169143575137\">\n<p id=\"fs-id1169143575139\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abc2208ce603280218658ad9f62b3ae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"226\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145777437\">\n<p id=\"fs-id1169145777439\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87f7da77b70157b4811f2a220af9dfd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#46;&#53;&#44;&#50;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> with a radius of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-153f0df57342ed9e1599f5446c7c0be7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147725203\" data-alt=\"This graph shows circle with center at (1.5, 2.5) and a radius of 0.5\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_321_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (1.5, 2.5) and a radius of 0.5\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145579182\">\n<div data-type=\"problem\" id=\"fs-id1169145579184\">\n<p id=\"fs-id1169145579187\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c4afa621cf5d1dfba6b0ee3c987674c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"177\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145731596\">\n<div data-type=\"problem\" id=\"fs-id1169145731598\">\n<p id=\"fs-id1169147741233\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-358bac6ea7b08a21b71ed22199d52e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147707766\">\n<p id=\"fs-id1169147707768\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 8.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147770825\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 8.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_323_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 8.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145578236\">\n<div data-type=\"problem\" id=\"fs-id1169145578238\">\n<p id=\"fs-id1169145578240\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4846d3ba91620499c56c560ff0529ae1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145577421\">\n<div data-type=\"problem\" id=\"fs-id1169145660340\">\n<p id=\"fs-id1169145660342\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-172623deb58008138833d8d1606ec77f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145733520\">\n<p id=\"fs-id1169145733522\"><span class=\"token\">\u24d0<\/span> The circle is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> with a radius of 2.<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148133834\" data-alt=\"This graph shows circle with center at (0, 0) and a radius of 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_325_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, 0) and a radius of 2.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147854265\">\n<div data-type=\"problem\" id=\"fs-id1169147854268\">\n<p id=\"fs-id1169147854270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e310edc8370ce421ac9e136a23d21edc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147965199\">In the following exercises, <span class=\"token\">\u24d0<\/span> identify the center and radius and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143575851\">\n<div data-type=\"problem\" id=\"fs-id1169143575853\">\n<p id=\"fs-id1169148081242\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19ef381ddc9972ad7e8dea3d2c35fffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#54;&#121;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147967370\">\n<p id=\"fs-id1169147967372\"><span class=\"token\">\u24d0<\/span> Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8eabd9bbf9ec5efccfce74c0bf77f00c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> radius: 1<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147875802\" data-alt=\"This graph shows circle with center at (negative 1, negative 3) and a radius of 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_327_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 1, negative 3) and a radius of 1.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143579426\">\n<div data-type=\"problem\" id=\"fs-id1169143579428\">\n<p id=\"fs-id1169143579430\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-931ce0c76f78044a842dcaef4cc3486f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#56;&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147864093\">\n<div data-type=\"problem\" id=\"fs-id1169147864095\">\n<p id=\"fs-id1169147864098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc73afbac658af3f7fb1ab28608a4ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#48;&#121;&#45;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"209\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145730625\">\n<p id=\"fs-id1169145730628\"><span class=\"token\">\u24d0<\/span> Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90bc5598050b450e3f413cd73afce445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> radius: 6<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147758639\" data-alt=\"This graph shows circle with center at (2, negative 5) and a radius of 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_329_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (2, negative 5) and a radius of 6.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148063172\">\n<div data-type=\"problem\" id=\"fs-id1169147804002\">\n<p id=\"fs-id1169147804005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2caeda2cd6c670bd4e7b996a457d092f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#49;&#52;&#121;&#43;&#50;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145663824\">\n<div data-type=\"problem\" id=\"fs-id1169145663826\">\n<p id=\"fs-id1169147856895\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e8dfa2b23795590c2ffc4449fe93348_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#121;&#43;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"160\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147960756\">\n<p id=\"fs-id1169147960758\"><span class=\"token\">\u24d0<\/span> Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3228e365cc5c34a8d1ac63acea4d0408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> radius: 2<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147879986\" data-alt=\"This graph shows circle with center at (0, negative 3) and a radius of 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_331_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (0, negative 3) and a radius of 2.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147805025\">\n<div data-type=\"problem\" id=\"fs-id1169147741808\">\n<p id=\"fs-id1169147741810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b24d322c891492f051c9b1200969fabb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148231447\">\n<div data-type=\"problem\" id=\"fs-id1169143580427\">\n<p id=\"fs-id1169143580429\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef6b895e514f7dc2587c74e22b6ad1cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147856831\">\n<p id=\"fs-id1169147856833\"><span class=\"token\">\u24d0<\/span> Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4050f6611d38892e64c174797e0a0e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> radius: = 2 <span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147794371\" data-alt=\"This graph shows circle with center at (negative 2, 0) and a radius of 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_333_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center at (negative 2, 0) and a radius of 2.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145493979\">\n<div data-type=\"problem\" id=\"fs-id1169145605845\">\n<p id=\"fs-id1169145605847\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-188c67c7348e573d0ff9990627e7f763_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#49;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"178\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169147873338\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169147764974\">\n<div data-type=\"problem\" id=\"fs-id1169147864549\">\n<p id=\"fs-id1169147864551\">Explain the relationship between the distance formula and the equation of a circle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143575555\">\n<p id=\"fs-id1169143575557\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147878929\">\n<div data-type=\"problem\" id=\"fs-id1169147878931\">\n<p id=\"fs-id1169145731327\">Is a circle a function? Explain why or why not.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148208053\">\n<div data-type=\"problem\" id=\"fs-id1169147965394\">\n<p id=\"fs-id1169147965396\">In your own words, state the definition of a circle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143349074\">\n<p id=\"fs-id1169143349076\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148072593\">\n<div data-type=\"problem\" id=\"fs-id1169148072595\">\n<p id=\"fs-id1169147830310\">In your own words, explain the steps you would take to change the general form of the equation of a circle to the standard form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169145499937\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1169147851532\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147834300\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_01_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<p id=\"fs-id1169147708321\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\n<p id=\"fs-id1169145644689\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\n<p id=\"fs-id1169147978241\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\n<p id=\"fs-id1169143575150\">\u2026no &#8211; I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1169145731792\">\n<dt>circle<\/dt>\n<dd id=\"fs-id1169147879662\">A circle is all points in a plane that are a fixed distance from a fixed point in the plane.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15310","chapter","type-chapter","status-publish","hentry"],"part":15253,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15310","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15310\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/15253"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15310\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15310"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15310"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15310"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}