45-45-90 Right Triangle

Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations.

• the special right triangle called the 45°-45°-90° triangle.
• how to solve problems involving the 45°-45°-90° right triangle

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 . Note that a 45°- 45°- 90° triangle is an isosceles right triangle.

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

Solving problems with 45°- 45°- 90° triangles

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.

Step 2: 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.  

The following videos show more examples of 45-45-90 triangles.

How to find the length of a leg or hypotenuse in a 45-45-90 triangle using the Pythagorean Theorem.

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