Question

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

Answer 1

Let's call the legs
`l_1` and
`l_2`,
`l_1` being the shortest, and the hypotenuse
`h`.

The first sentence translates to

`l_2 = 5l_1+19`

The second sentence translates to

`h = 5l_1+20`

So, we can express all sides in terms of
`l_1` and substitute the expressions in the Pythagorean threorem:

`h^2 = l_1^2+l_2^2 \to (5l_1+20)^2 = l_1^2 + (5l_1+19)^2`

Expand the squares:

`25 l_1^2 + 200 l_1 + 400 = l_1^2 + 25 l_1^2 + 190 l_1 + 361`

Bring everything to the left hand side and simplify:

`l_1^2 - 10 l_1 - 39 = 0`

This equation has solutions
`l_1 = -3` and
`l_1 = 13`.

We can't accept negative lengths, so the answer is
`l_1 = 13`.

From here, it's easy to get the other sides:

`l_2 = 5l_1+19 = 5 \cdot 13 + 19 = 84`

`h = 5l_1+20 = 5 \cdot 13 + 20 = 85`