<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

3.7: Pythagorean Theorem, Part 2: Applications & Triples

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Identify common Pythagorean triples.

Using Pythagorean Triples

Review the example problems from the previous lesson.

This is the diagram from Example 1:

In Example 1, the sides of the triangle are 3, 4, and 5. This combination of numbers is referred to as a Pythagorean triple. A Pythagorean triple is three integers (whole numbers with no decimal or fraction part) that make the Pythagorean Theorem true.

  • A Pythagorean triple is a group of three _____________________________ that satisfy the Pythagorean Theorem.

Throughout this chapter, you will learn other Pythagorean triples as well.

This is the diagram from Example 2:

Using the Pythagorean Theorem equation \begin{align*}a^2 + b^2 = c^2\end{align*}, and letting \begin{align*}a = 6\end{align*} and \begin{align*}c = 10\end{align*}, we calculated that \begin{align*}b = 8\end{align*} inches.

The triangle in Example 2 is proportional to the same ratio of 3 : 4 : 5. If you divide the lengths of the triangle (6, 8, and 10) by 2, you find the same proportion — 3 : 4 : 5 (because \begin{align*}6 \div 2 = 3, 8 \div 2 = 4\end{align*}, and \begin{align*}10 \div 2 = 5\end{align*}).

Whenever you find a Pythagorean triple, you can apply these ratios with greater factors as well.

Finally, look at the side lengths of the triangle in Example 3:

The two legs are 5 cm and 12 cm and the length of the missing side (the hypotenuse) is 13 cm. The side lengths make a ratio of 5 : 12 : 13. This, too, is a Pythagorean triple. You can infer that this ratio, multiplied by greater factors, will also yield numbers that satisfy the Pythagorean Theorem.

There are infinitely many Pythagorean triples, but a few of the most common ones and their multiples are in the chart below:

Pythagorean triple \begin{align*}\times 2\end{align*} \begin{align*}\times 3\end{align*} \begin{align*}\times 4\end{align*}
3 – 4 – 5 6 – 8 – 10 9 – 12 – 15 12 – 16 – 20
5 – 12 – 13 10 – 24 – 26 15 – 36 – 39 20 – 48 – 52
7 – 24 – 25 14 – 48 – 50 21 – 72 – 75 28 – 96 – 100
8 – 15 – 17 16 – 30 – 34 24 – 45 – 51 32 – 60 – 68

Reading Check:

1. What is a Pythagorean triple?

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

2. Which of the following is NOT a Pythagorean triple? Show your work.

a. 15 – 36 – 39

b. 15 – 20 – 25

c. 16 – 30 – 35

d. 25 – 60 – 65

3. Give 2 examples of Pythagorean triples that are NOT in the chart above.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

4. Why is it helpful to know common Pythagorean triples?

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

Image Attributions

Show Hide Details
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the section. Click Customize to make your own copy.
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original