# The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by 2 consecutive even integers. How do you find the lengths of the two legs?

Jan 4, 2016

$6 , 8$

#### Explanation:

The first thing to tackle here is how to express "two consecutive even integers" algebraically.

$2 x$ will give an even integer if $x$ is also an integer. The next even integer, following $2 x$, would be $2 x + 2$. We can use these as the lengths of our legs, but must remember that this will only hold true if $x$ is a (positive) integer.

Apply the Pythagorean theorem:

${\left(2 x\right)}^{2} + {\left(2 x + 2\right)}^{2} = {10}^{2}$

$4 {x}^{2} + 4 {x}^{2} + 8 x + 4 = 100$

$8 {x}^{2} + 8 x - 96 = 0$

${x}^{2} + x - 12 = 0$

$\left(x + 4\right) \left(x - 3\right) = 0$

$x = - 4 , 3$

Thus, $x = 3$ since the side lengths of the triangle can't be negative.

The legs are

$2 x \Rightarrow 6$
$2 x + 2 \Rightarrow 8$
$\text{hypotenuse} \Rightarrow 10$

A more intuitive way to do this problem is to recognize that a $6 , 8 , 10$ triangle is just twice the size of the fundamental $3 , 4 , 5$ right triangle.