3.2 Midpoint and Distance Between Points

Terrance Berg

Chapter 3: Graphing

3.2 Midpoint and Distance Between Points

Finding the Distance Between Two Points

The logic used to find the distance between two data points on a graph involves the construction of a right triangle using the two data points and the Pythagorean theorem $(a^2 + b^2 = c^2)$ to find the distance.

To do this for the two data points $(x_1, y_1)$ and $(x_2, y_2)$ , the distance between these two points $(d)$ will be found using $\Delta x = x_2 - x_1$ and $\Delta y = y_2 - y_1.$

Using the Pythagorean theorem, this will end up looking like:

$d^2 = \Delta x^2 + \Delta y^2$

The distance between (x1, y1) and (x2, y2) is the length of the hypotenuse.

or, in expanded form:

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

Delta x = x2 minus x1 (bottom leg). Delta y = y2 minus y1 (right leg).

On graph paper, this looks like the following. For this illustration, both $\Delta x$ and $\Delta y$ are 7 units long, making the distance $d^2 = 7^2 + 7^2$ or $d^2 = 98$ .

Triangle d sup 2, delta y sup 2, delta x sup 2

The square root of 98 is approximately 9.899 units long.

Example 3.2.1

Find the distance between the points $(-6,-4)$ and $(6, 5)$ .

Start by identifying which are the two data points $(x_1, y_1)$ and $(x_2, y_2)$ . Let $(x_1, y_1)$ be $(-6,-4)$ and $(x_2, y_2)$ be $(6, 5)$ .

Now:

$\Delta x^2 = (x_2 - x_1)^2$ or $[6 - (-6)]^2$ and $\Delta y^2 = (y_2 - y_1)^2$ or $[5 - (-4)]^2$ .

This means that

$d^2 = [6 - (-6)]^2 + [5 - (-4)]^2$

or

$d^2 = [12]^2 + [9]^2$

which reduces to

$d^2 = 144 + 81$

or

$d^2 = 225$

Taking the square root, the result is $d = 15$ .

Finding the Midway Between Two Points (Midpoint)

The logic used to find the midpoint between two data points $(x_1, y_1)$ and $(x_2, y_2)$ on a graph involves finding the average values of the $x$ data points $(x_1, x_2)$ and the of the $y$ data points $(y_1, y_2)$ . The averages are found by adding both data points together and dividing them by $2$ .