3.7: Pythagorean Theorem, Part 2: Applications & Triples
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.
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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*}
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