The length of the longer leg of a right triangle is 20 inches more than twice the length of the shorter leg. The length of the hypotenuse is 22 inches more than twice the length of the shorter leg. Find the side lengths of the triangle.

Step-by-step explanation:

Let x represent the length of the hypotenuse.

Let y represent the length of the the shorter leg.

Let z represent the length of the longer leg.

The length of the longer leg of a right triangle is 20 inches more than twice the length of the shorter leg. It means that

z = 2y + 20

The length of the hypotenuse is 22 inches more than twice the length of the shorter leg. It means that

x = 2y + 22

We would apply Pythagoras theorem which is expressed as

Hypotenuse² = opposite side² + adjacent side²

Therefore,

(2y + 22) ² = y² + (2y + 20) ²

(2y + 22) (2y + 22) = y² + (2y + 20) (2y + 20)

= 4y² + 44y + 44y + 484 = y² + 4y² + 40y + 40y + 400

4y² + 88y + 484 = y² + 4y² + 80y + 400

4y² + 88y + 484 = 5y² + 80y + 400

5y² - 4y² + 80y - 88y + 400 - 484

y² - 8y - 84 = 0

y² + 6y - 14y - 84 = 0

y (y + 6) - 14 (y + 6)

y - 14 = 0 or y + 6 = 0

y = 14 or y = - 6

The shorter length cannot be negative hence y = 14 inches

The length of the shorter side is 14 inches.

The length of the hypotenuse is

(2 * 14) + 22 = 50 inches

The length of the longer side is

(2 * 14) + 20 = 48 inches