Geometry question

Question

The altitude of an equilateral triangle measures 8 square root 3 units. Given this, determine the perimeter of the equilateral triangle in exact form.

Answer 1

If you think about an equilateral triangle, you know that each angle is 60°. If you draw in the altitude into the equilateral triangle, you will divide the triangle into two congruent triangles. Think about the angles in these two triangles. What are they?

You know that the altitude creates a right angle at the base and in an equilateral triangle, the altitude bisects the angle at the vertex.

Knowing this, you will have a 30° angle at the top with a 90° on the bottom and the 60° angle from the original equilateral triangle. Does this give you a special triangle? Yes, a 30-60-90 triangle.

If the longer leg (in this case the altitude) is 8*√3 than what is the short leg and the hypotenuse?

The sides of a 30-60-90 triangle are given by short leg x, long leg x√3 and hypotenuse 2x.

With the long leg given as 8√3, what is x? 8. You should be able to find the hypotenuse from the given formulas for a 30-60-90 triangle. Once you have the hypotenuse you can get the perimeter.

Hope this helped.

Answer 2

To understand this problem, it's good to remember about the 30-60-90 special right triangle.

Let "s" be the "short" side of a 30-60-90 triangle. Then the hypotenuse is twice that length (2s) and the other leg is s√3 like this:

An equilateral triangle has a 30-60-90 in it like this:

so if h = 8√3, then x would be 8 which makes each side of the equilateral triangle 16 units long. If each side is 16, then the perimeter is 16•3 or 48 units long.