Doug C. answered • 12/03/21

Math Tutor with Reputation to make difficult concepts understandable

Here are some theorems from Geometry,

In a 30-60-90 right triangle the hypotenuse is twice the short leg.

The long leg is √ 3 times the short leg.

And of course the short leg is opposite the 30 degree angle and the long leg is opposite the 60 degree angle. If the side opposite 30 degrees was 6 (for example, then the hypotenuse would be 12 and the long leg would be 6√3. Apply the same pattern with your problem.

These theorems can be proved by dropping an altitude from one of the angles in an equilateral triangle (where each angle is 60 degrees) to its opposite side. That altitude creates two congruent triangles (provable by HL). That means the vertex angle is bisected, so the congruent triangles are each 30-60-90. Also the side opposite the bisected angle is bisected too. So the side opposite 30 degrees is 1/2 the hypotenuse. If we let x = length of side of the equilateral triangle and h the altitude, applying the Pythagorean theorem gives (x/2)^{2} + h^{2} = x^{2}.

Solve for h as follows:

x^{2}/4 + h^{2} = 4x^{2}/4

h^{2}=3x^{2}/4

h = x√3/2

Which happens to be √3 times the short leg (which is x/2).