45-45-90 Right Triangle

There are three types of special triangles: 3-4-5 triangles, 45°-45°-90° triangles and 30°-60°-90° triangles. In this lesson we will study 45°-45°-90° triangles.

45º-45º-90º Triangles

A 45°- 45°- 90° triangle is a special right triangle whose angles are 45°, 45°and 90°. The lengths of the sides of a 45°- 45°- 90° triangle are in the ratio of .

A right triangle with two sides of equal lengths is a 45°- 45°- 90° triangle.

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches.

Solution:

Step 1: This is a right triangle with two equal sides so it must be a 45°- 45°- 90° triangle.

Step 2: You are given that the both the sides are 3. If the first and second value of the ratio  is 3 then the length of the third side is

Answer: The length of the hypotenuse is  inches.

You can also recognize a 45°- 45°- 90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is 45° then it must be a 45°- 45°- 90° special right triangle.

A right triangle with a 45° angle must be a 45°- 45°- 90° special right triangle.

Example 2:

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is inches and one of the angles is 45°.

Solution:

Step 1: This is a right triangle with a 45°so it must be a 45°- 45°- 90° triangle.

You are given that the hypotenuse is . If the third value of the ratio  is then the lengths of the other two sides must 4.

Answer: The lengths of the two sides are both 4 inches.