Question

The shorter leg of a right triangle is 6 m shorter than the longer leg. The hypotenuse is 6 m longer than the longer leg. What are the side lengths of the triangle?

Answer 1

Length of the longer side = `24m`
Length of the shorter side = `18m`
Length of the hypotenuse = `30m`

Explanation:

From the data given, we can write:

Length of the longer side = `x`
Length of the shorter side = `x-6`
Length of the hypotenuse = `x+6`

The Pythagorean Theorem states the the square of the hypotenuse is equal to the sum of the squares of the other two sides. Hence.

`x^2+(x-6)^2=(x+6)^2`

`x^2+x^2-12x+36=x^2+12x+36`

Cancel equivalent terms on each side.

`x^2+cancelx^2-12x+cancel36=cancelx^2+12x+cancel36`

`x^2-12x=12x`

Subtract `12x` from each side.

`x^2-24x=0`

Factorise.

`x(x-24)=0`

`x=0` or `x-24=0`

`x=0` or `x=24`

Since the length has to be a positive integer, the length of the longer side is `24m`.

Hence:
Length of the longer side = `24m`
Length of the shorter side = `18m`
Length of the hypotenuse = `30m`

To verify, we apply the Pythagorean Theorem:

`24^2+18^2=30^2`

`576+324=900`

Since the two sides are equal, the calculations have been verified.