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Learning Objectives

- Recognize Special Right Triangles.
- Use the special right triangle rations to solve special right triangles.

### 30-60-90 Right Triangles

Hypotenuse equals twice the smallest leg, while the larger leg is \(\sqrt{3}\) times the smallest.

One of the two special right triangles is called a **30-60-90 triangle**, after its three angles.

** 30-60-90 Theorem:** If a triangle has angle measures \(30^{\circ}\), \(60^{\circ}\) and \(90^{\circ}\), then the sides are in the ratio \(x: x\sqrt{3}:2x\).

The shorter leg is always \(x\), the longer leg is always \(x\sqrt{3}\), and the **hypotenuse** is always \(2x\). If you ever forget these theorems, you can still use the **Pythagorean Theorem**.

What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the lengths of its other sides?

Example \(\PageIndex{1}\)

Find the value of \(x\) and \(y\).

**Solution**

We are given the longer leg.

\(\begin{aligned} x\sqrt{3} &=12 \\ x&=12\sqrt{3}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}=12\dfrac{\sqrt{3}}{3}=4\sqrt{3} \\ &\text{The hypotenuse is} \\ y&=2(4\sqrt{3})=8\sqrt{3} \end{aligned}\)

Example \(\PageIndex{2}\)

Find the value of \(x\) and \(y\).

**Solution**

We are given the hypotenuse.

\(\begin{aligned} 2x&=16 \\ x&=8 \\ \text{The longer leg is} \\ y&=8\cdot \sqrt{3}&=8\sqrt{3} \end{aligned} \)

Example \(\PageIndex{3}\)

Find the length of the missing sides.

**Solution**

We are given the shorter leg. If \(x=5\), then the longer leg, \(b=5\sqrt{3}\), and the hypotenuse, \(c=2(5)=10\).

Example \(\PageIndex{4}\)

Find the length of the missing sides.

**Solution**

We are given the hypotenuse. \(2x=20\), so the shorter leg, \(f=\dfrac{20}{2}=10\), and the longer leg, \(g=10\sqrt{3}\).

Example \(\PageIndex{5}\)

A rectangle has sides 4 and \(4\sqrt{3}\). What is the length of the diagonal?

**Solution**

If you are not given a picture, draw one.

The two lengths are \(x\), \(x\sqrt{3}\), so the diagonal would be \(2x\), or \(2(4)=8\).

If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.

\(\begin{aligned} 4^2+(4\sqrt{3})^2&=d^2 \\ 16+48&=d^2 \\ d=\sqrt{64}&=8 \end{aligned}\)

**Review**

- In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
- In a 30-60-90 triangle, if the shorter leg is \(x\), then the longer leg is __________ and the hypotenuse is ___________.
- A rectangle has sides of length 6 and \(6\sqrt{3}\). What is the length of the diagonal?
- Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides

**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).

\(\Delta ABC\) 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 \(\PageIndex{6}\)

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.

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

Example \(\PageIndex{7}\)

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 \(\PageIndex{8}\)

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 \(\PageIndex{9}\)

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 \(\PageIndex{10}\)

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

**Solution**

Draw a picture.

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

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

### 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?

## Resources

## Vocabulary

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

30-60-90 Theorem | If a triangle has angle measures of 30, 60, and 90 degrees, then the sides are in the ratio \(x : x \sqrt{3} : 2x\) |

45-45-90 Theorem | For any isosceles right triangle, if the legs are x units long, the hypotenuse is always \(x\sqrt{2}\). |

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. |

Pythagorean Theorem | The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by \(a^2+b^2=c^2\), where a and b are legs of the triangle and c is the hypotenuse of the triangle. |

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

## Additional Resources

Interactive Element

Video: Solving Special Right Triangles

Activities: 30-60-90 Right Triangles Discussion Questions

Study Aids: Special Right Triangles Study Guide

Practice: 30-60-90 Right Triangles 45-45-90 Right Triangles

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