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

Let x represent the length of the shorter leg.

Longer leg: 5x +19
Hypotenuse: 5x +20

Pythagorean theorem:
.. x^2 +(5x +19)^2 = (5x +20)^2
.. x^2 +25x^2 +190x +19^2 = 25x^2 +200x +20^2 . . . . . . eliminate parentheses
.. x^2 -10x -39 = 0
.. (x -13)(x +3) = 0
.. x = 13 . . . . . . . . . . leg lengths must be positive

The side length of the triangle are 13, 84, 85 feet.

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