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# Special Right Triangles

## What is a Special Right Triangle?

**There are certain right triangles with dimensions that make remembering the side lengths and angles very easy.** These are known as special right triangles. Special right triangles fall into two categories: angle-based and side-based. We will go over the common and useful angle-based and side-based triangles in this lesson.

### Angle-Based Special Right Triangles

The common angle-based special right triangles are:

45-45-90 Triangle

30-60-90 Triangle**The triangle name describes the three internal angles.** These triangles also have side length relationships that can be easily memorized. The image below shows all angle and side length relationships for the 45-45-90 and 30-60-90 triangles.

### Side-Based Special Right Triangles

The common side-based special right triangles are:

3-4-5 Triangle

5-12-13 Triangle**The triangle name describes the ratio of side lengths.** For example, a 3-4-5 triangle could have side lengths of 6-8-10 since they have a 3-4-5 ratio. The image below shows all side length and angle relationships for the 3-4-5 and 5-12-13 triangles.

## How to Solve Special Right Triangles

The reason for memorizing the special right triangles is that it allows us to quickly determine a missing side length or angle. **The first step in solving any special right triangle problem is to identify what type of triangle it is.**

Once the type of special right triangle has been identified, we are usually able to determine the missing side length or angle. Take a look at the practice problems below to see how we do this.

### Special Right Triangle Practice Problems

**Problem 1:**

A 45-45-90 triangle has two sides with a length of 10. What is the 3rd side length?

Solution:

The 45-45-90 triangle relationship tells us that the hypotenuse is square root of 2 times the leg. Since the leg is 10, the hypotenuse/**3rd side length is **.

**Problem 2:**

A triangle has side two internal angles of 30° and 90°, and two side lengths of 5 and . What is the 3rd side length?

Solution:

This must be a 30-60-90 triangle because of the two given angles. The 30-60-90 relationship tells us that the side lengths are *a*, *2a*, and *a√3*. We can see from the two given sides that a = 5 and we are missing the *2a* side. So, **the 3rd side length is 2·5 = 10**.

**Problem 3:**

A triangle has side lengths of 20 and 48. What is the 3rd side length?

Solution:

Let’s figure out which side-based special right triangle this is. First, reduce the side lengths by a common denominator. 20/4 = 5 and 48/4 = 12, so this must be a 5-12-13 triangle. 13·4 = 52, so **the 3rd side length is 52**.

**Problem 4:**

A triangle has side lengths of 21 and 28. What is the 3rd side length?

Solution:

Let’s figure out which side-based special right triangle this is. First, reduce the side lengths by a common denominator. 21/7 = 3 and 28/7 = 4, so this must be a 3-4-5 triangle. 5·7 = 35, so **the 3rd side length is 35**.