4.42: 45-45-90 Right Triangles

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).

Figure \(\PageIndex{1}\)

\(\Delta ABC\) 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 \(x\).

Figure \(\PageIndex{2}\)

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.

Figure \(\PageIndex{3}\)

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.

Figure \(\PageIndex{4}\)

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

Figure \(\PageIndex{5}\)

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

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 \(x\), 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.

5. 

   Figure \(\PageIndex{6}\)

6. 

   Figure \(\PageIndex{7}\)

7. 

   Figure \(\PageIndex{8}\)

8. 

   Figure \(\PageIndex{9}\)

9. 

   Figure \(\PageIndex{10}\)

10. 

    Figure \(\PageIndex{11}\)

11. 

    Figure \(\PageIndex{12}\)

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