Question

The longer leg of a right triangle is 1 m longer than the shorter leg. The hypotenuse is 9 m longer than the shorter leg Find the side lengths of the triangl

Answer 1

20 m , 21 m and 29 m.

Explanation:

Let the length of the shorter leg be x m. Then the longer leg = x + 1 m and the hypotenuse = x + 9 m. So, by Pythagoras:

(x + 9)^2 = x^2 + (x + 1)^2

x^2 + 18x + 81 = x^2 + x^2 + 2x + 1

18x + 81 = x^2 + 2x + 1

x^2 - 16x - 80 = 0

(x - 20)(x + 4) = 0

x = 20, -4 (as we are dealing with lengths of sides we ignore the negative root).

So the sides have length 20, 21 and 29 (answer).

Answer 2

Let
`x` be the length of the shorter leg.

The other leg is 1m longer, so its length is
`x+1`

The hypothenuse is 9m longer, so its length is
`x+9`

The pythagorean theorem states that the sum of the squares of the legs is the square of the hypothenuse, so we have

`x^2+(x+1)^2=(x+9)^2`

Expanding the squares gives

`x^2+x^2+2x+1=x^2+18x+81`

Move all to the left hand side:

`x^2-16x-80=0`

This equation has solutions
`x=-4` and
`x=20`

We can't accept the first solution, because it would lead to the side lengths

`x=-4,\quad x+1=-3,\quad x+9=5`

And we can't have negative side lengths.

The other solution is fine, because it leads to the side lengths

`x=20,\quad x+1=21,\quad x+9=29`

So, the side lengths are 20 (shorter leg), 21 (longer leg), 29 (hypothenuse)