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# 45-45-90 Right Triangles

## Leg times sqrt(2) equals hypotenuse.

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45-45-90 Right Triangles

### 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).

ABC\begin{align*}\triangle ABC\end{align*} is a right triangle with mA=90\begin{align*}m \angle A = 90^\circ\end{align*}, AB¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯\begin{align*} \overline {AB} \cong \overline{AC}\end{align*} and mB=mC=45\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 x:x:x2\begin{align*}x:x:x \sqrt{2}\end{align*}. For any isosceles right triangle, the legs are x\begin{align*}x\end{align*} and the hypotenuse is always x2\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 x\begin{align*}x\end{align*}.

Use the x:x:x2\begin{align*}x:x:x \sqrt{2}\end{align*} ratio.

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

x2x=16=16222=1622=82\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 x\begin{align*}x\end{align*}, where x\begin{align*}x\end{align*} is the hypotenuse of a 45-45-90 triangle with leg lengths of 53\begin{align*}5\sqrt{3}\end{align*}.

Use the x:x:x2\begin{align*}x:x:x \sqrt{2}\end{align*} ratio.

x=532=56\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 x:x:x2\begin{align*}x:x:x \sqrt{2}\end{align*} ratio. TV=6\begin{align*}TV = 6\end{align*} because it is equal to ST\begin{align*}ST\end{align*}. So, SV=62=62\begin{align*}SV = 6 \cdot \sqrt{2} = 6 \sqrt{2}\end{align*}.

#### Example 4

Find the length of the missing side.

Use the x:x:x2\begin{align*}x:x:x \sqrt{2}\end{align*} ratio. AB=92\begin{align*}AB = 9 \sqrt{2}\end{align*} because it is equal to AC\begin{align*}AC\end{align*}. So, BC=922=92=18\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 10=s2\begin{align*}10=s \sqrt{2}\end{align*}.

s2s=10=10222=1022=52\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

1. In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
2. In an isosceles right triangle, if a leg is x\begin{align*}x\end{align*}, then the hypotenuse is __________.
3. A square has sides of length 15. What is the length of the diagonal?
4. 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.

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

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Color Highlighted Text Notes

### Vocabulary Language: English

45-45-90 Theorem

For any isosceles right triangle, if the legs are x units long, the hypotenuse is always x$\sqrt{2}$.

45-45-90 Triangle

A 45-45-90 triangle is a special right triangle with angles of $45^\circ$, $45^\circ$, and $90^\circ$.

Hypotenuse

The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.

Legs of a Right Triangle

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.

The $\sqrt{}$, or square root, sign.