Question

The legs of a right triangle have lengths of x + 4 and x + 7. The hypotenuse length is 3x. How do you find the perimeter of the triangle?

Answer 1

`36`

Explanation:

The perimeter is equal to the sum of the sides, so the perimeter is:

`(x+4)+(x+7)+3x=5x+11`

However, we can use the Pythagorean theorem to determine the value of `x` since this is a right triangle.

`a^2+b^2+c^2`

where `a,b` are legs and `c` is the hypotenuse.

Plug in the known side values.

`(x+4)^2+(x+7)^2=(3x)^2`

Distribute and solve.

`x^2+8x+16+x^2+14x+49=9x^2`

`2x^2+22x+65=9x^2`

`0=7x^2-22x-65`

Factor the quadratic (or use the quadratic formula).

`0=7x^2-35x+13x-65`

`0=7x(x-5)+13(x-5)`

`0=(7x+13)(x-5)`

`x=-13/7,5`

Only `x=5` is valid here, since the hypotenuse's length would be negative if `x=-13/7`.

Since `x=5`, and the perimeter is `5x+11`, the perimeter is:

`5(5)+11=36`