finding the length of the right triangle

Question

The length of the longer leg of a right triangle is 19m

 more than five times the length of the shorter leg. The length of the hypotenuse is 20m

 more than five times the length of the shorter leg. Find the side lengths of the triangle. 

Answer 1

Hi Princess,

We don't know the length of the short leg, so let's call it x.

The longer leg of the right triangle is 19 m more than five times the length of the short leg: 19 + 5x

The hypotenuse is 20 m more than five times the length of the short leg: 20 + 5x

Using the Pythagorean Theorem: a² + b² = c²

x² + (19 + 5x)² = (20 + 5x)²

x² + (19 + 5x)(19 + 5x) = (20 + 5x)(20 + 5x)

x² + 361 + 95x + 95x + 25x² = 400 + 100x + 100x + 25x²

26x² + 190x + 361 = 25x² + 200x + 400

x² - 10x - 39 = 0

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

x - 13 = 0 -> x = 13

x + 3 = 0 -> x = -3 (we cannot have a negative length, so we can discard this value)

Since the short leg is x = 13m, the longer leg is 19 + 5(13) = 84m, and the hypotenuse is 20 + 5(13) = 85m.

Hope this helps! :)