# 4.42: 45-45-90 Right Triangles

- Page ID
- 4983

Leg times \(\sqrt{2}\) equals hypotenuse.

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).

\(\Delta ABC\) is a right triangle with \(m\angle A=90^{\circ}\), \(\overline{AB}\cong \overline{AC}\) and \(m\angle B=m\angle C=45^{\circ}\).

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

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?

Example \(\PageIndex{1}\)

Find the length of \(x\).

**Solution**

Use the \(x:x:x\sqrt{2}\) ratio.

Here, we are given the hypotenuse. Solve for \(x\) in the ratio.

\(\begin{aligned} x\sqrt{2}&=16 \\ x&=\dfrac{16}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{16\sqrt{2}}{2}=8\sqrt{2}\end{aligned}\)

Example \(\PageIndex{2}\)

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

**Solution**

Use the \(x:x:x\sqrt{2}\) ratio.

\(x=5\sqrt{3}\cdot \sqrt{2}=5\sqrt{6}\)

Example \(\PageIndex{3}\)

Find the length of the missing side.

**Solution**

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

Example \(\PageIndex{4}\)

Find the length of the missing side.

**Solution**

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

Example \(\PageIndex{5}\)

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

**Solution**

We know half of a square is a 45-45-90 triangle, so \(10=s\sqrt{2}\).

\(\begin{aligned} s\sqrt{2}&=10 \\ s&=10\sqrt{2}\cdot \sqrt{2}\sqrt{2}=10\sqrt{2}2=5\sqrt{2}\end{aligned}\)

## Review

- In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
- In an isosceles right triangle, if a leg is \(x\), 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.

## Resources

## Vocabulary

Term | Definition |
---|---|

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. |

Radical |
The \(\sqrt\), or square root, sign. |

## Additional Resources

Interactive Element

Video: Solving Special Right Triangles

Activities: 45-45-90 Right Triangles Discussion Questions

Study Aids: Special Right Triangles Study Guide

Practice: 45-45-90 Right Triangles

Real World: Fighting the War on Drugs Using Geometry and Special Triangles