# 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?

Jan 3, 2016

$36$

#### Explanation:

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

$\left(x + 4\right) + \left(x + 7\right) + 3 x = 5 x + 11$

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.

${\left(x + 4\right)}^{2} + {\left(x + 7\right)}^{2} = {\left(3 x\right)}^{2}$

Distribute and solve.

${x}^{2} + 8 x + 16 + {x}^{2} + 14 x + 49 = 9 {x}^{2}$

$2 {x}^{2} + 22 x + 65 = 9 {x}^{2}$

$0 = 7 {x}^{2} - 22 x - 65$

Factor the quadratic (or use the quadratic formula).

$0 = 7 {x}^{2} - 35 x + 13 x - 65$

$0 = 7 x \left(x - 5\right) + 13 \left(x - 5\right)$

$0 = \left(7 x + 13\right) \left(x - 5\right)$

$x = - \frac{13}{7} , 5$

Only $x = 5$ is valid here, since the hypotenuse's length would be negative if $x = - \frac{13}{7}$.

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

$5 \left(5\right) + 11 = 36$