Question

If the altitude of an equilateral triangle is `8sqrt3`, what is the perimeter of the triangle?

Answer 1

The altitude of an equilateral triangle is the perpendicular line from a vertex to the opposite side.

This always forms a 30, 60, 90 triangle with the two triangles that form. (Try to prove this for "fun" Hint: HL = HL and CPCTC)

What do we know about 30-60-90 triangles? We know the relationship between angles and side lengths!

Since the altitude is opposite the 60 degree angle, the altitude is equal to `xsqrt3`.

Therefore, we can set `8sqrt3 = xsqrt3`

Clearly, `x = 8`

Since the side opposite the 30 degree angle is x, the other leg length would be `8`

Finally, we know that the hypotenuse is 2x, so the length would be `16`

We are trying to find the perimeter; we know that one side is 16.

Alas, it is an equilateral triangle, so all of the sides are length 16.

So we just add the lengths up, `16 + 16+16 = 48`