The length of the longer leg of a right triangle is 19 inches more than five times the length of the shorter leg. The length of the hypotenuse is 20 inches more than five times the length of the shorter leg. Find the side lengths of the triangle.

short leg: x

long leg: 5x + 19

hypotenuse: 5x + 20

(short leg) ² + (long leg) ² = hypotenuse

(x) ² + (5x + 19) ² = (5x + 20) ²

x² + 25x² + 190x + 361 = 25x² + 200x + 400

x² - 10x - 39 = 0 subtracted 25x², 200x, and 400 from both sides

(x - 13) (x + 3) = 0

x - 13 = 0, x + 3 = 0

x = 13 x = - 3 length cannot be negative so disregard - 3

long leg: 5x + 19 = 5 (13) + 19 = 65 + 19 = 84

hypotenuse: 5x + 20 = 5 (13) + 20 = 65 + 20 = 85

Answer: short leg = 13, long leg = 84, hypotenuse = 85