+383 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?
+500 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
x2 given a side length x, so we then have:
\(15^2 = 225\) in2 as the area of our square
