A square and a right triangle have equal perimeters. The legs of the right triangle are 20 inches and 15 inches. What is the area of the square, in square inches?

Paresh Mar 25, 2020

#1**0 **

Notice that the legs of the right triangle are in a ratio of 3 : 4 . If you remember anything about right triangles, you'll know the "3 - 4 -5" right triangle, which is a right triangle of side lengths 3,4, and 5(is there another way to phrase this ? lol). We can see that this right triangle of legs 20 and 15 inches is a "scaled up" version of the 3 4 5 right triangle, and is in fact, similar to the 3-4-5 triangle(by a ratio of 5). As such, we know that the hypotenuse of this triangle must be 5 * 5 = 25 inches long. Alternatively, we can just use pythagorean theorem to find the length of the hypotenuse of our triangle. Using pythagorean theorem, we have where *h* is the hypotenuse:

\(h^2 = 15^2 + 20^2 = 225 + 400 = 625\)

\(h = \sqrt{625} = 25\)

That means that the perimeter of said right triangle is:

\(15+20+25 = 60\)

Because the square and right triangle both have equal perimeters, we can name the side length of the square *x*

We can then write the equation:

\(4x = 60\), because the square has 4 sides of length x, which gives us its perimeter of x + x + x + x = 4x

Dividing by 4 on both sides, we get:

\(x = 15\)

The area of a square is equal to

x^{2} given a side length x, so we then have:

\(15^2 = 225\) in^{2} as the area of our square

jfan17 Mar 25, 2020