Question

What is the length of the altitude of an equilateral triangle whose perimeter is 15 inches?

Answer 1

`(5sqrt(3))/2 "inches"`

Explanation:

First off, since the perimeter of the equilateral triangle (meaning that all 3 sides have the same length) is 15 inches, each side has a length of 5 inches.

Now we must recognize that an altitude of a triangle spans from one vertex to the opposite side, creating a right angle at the point of intersection. Thus, drawing the altitude inside of the equilateral triangle will create two 30-60-90 right triangles with a hypotenuse of 5 inches.

These angles make sense because equilateral triangles have 3 60 degree angles and the remaining angle (we already have one right angle - 90 degrees - and a 60 degree angle) must be 30 degrees since the angles in any triangle must sum to 180 degrees. Now there are many ways to find the length of the altitude, and I will present the easiest way in my opinion.

We will be using the fact that a 30-60-90 right triangle has side lengths in the respective ratio of 1-`sqrt(3)`-2. And since the hypotenuse in question has a side length of 5, we can set up a ratio to find the side opposite the 60 degree angle. We'll call this length `alpha`.

`alpha/sqrt(3) = 5/2`
`color(red)(alpha = (5sqrt(3))/2 "inches")`