## 5 12 13 Triangle Lesson

### What is a 5 12 13 Triangle?

**The 5 12 13 triangle is an SSS special right triangle with the ratio between its side lengths as 5, 12, and 13.** It is a common Pythagorean triple that is worth memorizing to save time when dealing with right triangles. The other common SSS special right triangle is the 3 4 5 triangle.

### Using the 5 12 13 Ratio to our Advantage

A 5 12 13 triangle is considered a scalene triangle because all three of its sides have different lengths. If we come across a right triangle and two of the known sides are part of the 5 12 13 ratio, we can immediately determine that the third side will be the remaining number in the ratio. Note that the "13" side will always be the hypotenuse. **The 5 12 13 ratio is scalable and applies to any right triangle with sides that are any common multiple of the numbers 5, 12, and 13.**

### List of 5 12 13 Triangles

The following triangle side lengths are common multiples of the 5 12 13 ratio and are all considered 5 12 13 triangles. It is not necessary to memorize this list, but it is very useful to understand that 5 12 13 is a ratio that may manifest itself as smaller or larger numbers than the ratio itself.

- 2.5 6 6.5
- 5 12 13
- 7.5 18 19.5
- 15 36 39

### 5 12 13 Triangle Angles

A 5 12 13 triangle contains the following internal angles in degrees:

**22.6°, 67.4°, 90°**

And in radians:

**0.39, 1.18, and 1.57**

It is not expected to memorize these. We can use the soh cah toa rule and inverse trigonometric functions to solve for the angles since we know the side lengths. **See the image below for a visual of these angles.**

### What is the Area of a 5 12 13 Triangle?

Calculating the area of a 5 12 13 triangle, we get **A = ( ^{1}⁄_{2})(5)(12) = 30**. If the triangle conforms to the 5 12 13 ratio but is scaled, i.e. a 2.5 6 6.5 triangle, we still use the two shorter sides as the base and height. In this case, the area would be (

^{1}⁄

_{2})(2.5)(6) = 7.5.

### 5 12 13 Triangle Example Problem 1

Find the third side length of the 5 12 13 right triangle that has side lengths of 39 and 15.

Solution:

- Since we know this is a 5 12 13 triangle scaled from the ratio by an unknown factor, we must determine which two sides are given.

39/15 = 2.6

12/5 = 2.5

13/5 = 2.6 - Therefore, we have been given the "13" and "5" sides. We will now multiply by a ratio to get the "12" side.
- (
^{12}⁄_{13})(39) = 36 **The third side length is 36**.

### 5 12 13 Triangle Example Problem 2

Which of the following sets of side lengths constitute a 5 12 13 triangle?

- Triangle 1: 4, 9, 11.5
- Triangle 2: 10, 24, 26
- Triangle 3: 6, 13, 14
- Triangle 4: 20, 48, 52

Solution:

The side lengths of **triangles 2 and 4 constitute a 5 12 13 triangle**.