13 Origin of a 30-60-90 Triangle

Consider the Equilateral Triangle ABC where all sides have a measure of 2 units.
Since all sides have the same measure, then the measure of each angle is 180° divided by 3 = 60°

We then take ABC and cut it in two by drawing the angle bisector. From geometry, we know that in an equilateral triangle the angle bisector is also the altitude and the median. This really means that when we draw any one of these, angle A will be divided in 2 equal parts, BC will be divided in 2 equal parts, and AD will intersect BC at 90°.

In ABC, we can use The Theorem of Pythagoras to find the value of AD. Quickly, this means 2² – 1² = AD² or 3 = AD² giving Ö3 = AD. Thus, in a 30-60-90 Triangle, the side opposite the right angle (hypotenuse) is 2, the side opposite the smallest angle is 1, and the side opposite the 60° angle is Ö3.

We also note that the hypotenuse is always double the smallest side in a 30-60-90
Triangle.