# How do you find the legs in a 45-45-90 triangle when its hypotenuse is 11?

Mar 23, 2018

The two legs of the right isosceles triangle are both $\frac{11 \sqrt{2}}{2} \approx 7.78$.

#### Explanation:

Since two of the angles in this triangle are $45$ degrees and it has a $90$ degree angle, it is a right isosceles triangle. An isosceles triangle has two sides the same, which have to be the two legs in this triangle because a triangle can not have two hypotenuses.
According to the Pythagorean Theorem:
${a}^{2} + {b}^{2} = {c}^{2}$
where $a$ and $b$ are the legs and $c$ is the hypotenuse.
Since the two legs in this right triangle are the same, the formula can be altered to be:
${a}^{2} + {a}^{2} = {c}^{2}$
$2 {a}^{2} = {c}^{2}$

Plugging in $10$ for $c$:
$2 {a}^{2} = {11}^{2}$
$2 {a}^{2} = 121$
${a}^{2} = \frac{121}{2}$
$a = \sqrt{\frac{121}{2}}$
$a = \frac{\sqrt{121}}{\sqrt{2}}$
$a = \frac{11}{\sqrt{2}}$
$a = \frac{11 \sqrt{2}}{2} \approx 7.78$