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# 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 a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}, and letting a=6\begin{align*}a = 6\end{align*} and c=10\begin{align*}c = 10\end{align*}, we calculated that b=8\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 6÷2=3,8÷2=4\begin{align*}6 \div 2 = 3, 8 \div 2 = 4\end{align*}, and 10÷2=5\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 ×2\begin{align*}\times 2\end{align*} ×3\begin{align*}\times 3\end{align*} ×4\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

1. What is a Pythagorean triple?

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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.

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

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

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

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

8 , 9 , 10

## Date Created:

Feb 23, 2012

May 12, 2014
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