### 45-45-90 Right Triangles

A right triangle with congruent legs and acute angles is an **Isosceles Right Triangle**. This triangle is also called a 45-45-90 triangle (named after the angle measures).

\begin{align*}\triangle ABC\end{align*} is a right triangle with \begin{align*}m \angle A = 90^\circ\end{align*}, \begin{align*} \overline {AB} \cong \overline{AC}\end{align*} and \begin{align*}m \angle B = m \angle C = 45^\circ\end{align*}.

**45-45-90 Theorem:** If a right triangle is isosceles, then its sides are in the ratio \begin{align*}x:x:x \sqrt{2}\end{align*}. For any isosceles right triangle, the legs are \begin{align*}x\end{align*} and the hypotenuse is always \begin{align*}x \sqrt{2}\end{align*}.

What if you were given an isosceles right triangle and the length of one of its sides? How could you figure out the lengths of its other sides?

### Examples

#### Example 1

Find the length of \begin{align*}x\end{align*}.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*} ratio.

Here, we are given the hypotenuse. Solve for \begin{align*}x\end{align*} in the ratio.

\begin{align*}x \sqrt{2} &= 16\\ x &= \frac{16}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{16 \sqrt{2}}{2} = 8 \sqrt{2}\end{align*}

#### Example 2

Find the length of \begin{align*}x\end{align*}, where \begin{align*}x\end{align*} is the hypotenuse of a 45-45-90 triangle with leg lengths of \begin{align*}5\sqrt{3}\end{align*}.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*} ratio.

\begin{align*}x=5\sqrt{3}\cdot \sqrt{2}=5\sqrt{6}\end{align*}

#### Example 3

Find the length of the missing side.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*} ratio. \begin{align*}TV = 6\end{align*} because it is equal to \begin{align*}ST\end{align*}. So, \begin{align*}SV = 6 \cdot \sqrt{2} = 6 \sqrt{2}\end{align*}.

#### Example 4

Find the length of the missing side.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*} ratio. \begin{align*}AB = 9 \sqrt{2}\end{align*} because it is equal to \begin{align*}AC\end{align*}. So, \begin{align*}BC = 9 \sqrt{2} \cdot \sqrt{2} = 9 \cdot 2 = 18\end{align*}.

#### Example 5

A square has a diagonal with length 10, what are the lengths of the sides?

Draw a picture.

We know half of a square is a 45-45-90 triangle, so \begin{align*}10=s \sqrt{2}\end{align*}.

\begin{align*}s \sqrt{2} &= 10\\ s &= \frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}= \frac{10 \sqrt{2}}{2}=5 \sqrt{2}\end{align*}

### Review

- In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
- In an isosceles right triangle, if a leg is \begin{align*}x\end{align*}, then the hypotenuse is __________.
- A square has sides of length 15. What is the length of the diagonal?
- A square’s diagonal is 22. What is the length of each side?

For questions 5-11, find the lengths of the missing sides. Simplify all radicals.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.5.