# A 45-45-90 triangle has a hypotenuse of length 14 units. What is the length of one of the legs?

Nov 14, 2015

One side length is equal to $\sqrt{98}$.

#### Explanation:

We can start off by establishing that this triangle is a right, isosceles triangle. We can say this because of the converse of the isosceles triangle theorem, which states if two angles are congruent, then then 2 sides are congruent.

What we can conclude from the previous statement, is that the two legs are equal length. Therefore, we can use a modified version of the Pythagorean Theorem. That is, since $a \cong b$, we can substitute $a$ in for $b$. That makes our equation ${a}^{2} + {a}^{2} = {c}^{2}$.

We know that the hypotenuse is $14$, so we can plug that into the equation. ${a}^{2} + {a}^{2} = {14}^{2} \implies 2 {a}^{2} = 196$. We can divide each side by 2, and square root each side to rid of the exponent.

That leaves us with $a = \sqrt{98} .$