# 1.4: Pythagorean Triples

Integer triples that make right triangles.

While working as an architect's assistant, you're asked to utilize your knowledge of the Pythagorean Theorem to determine if the lengths of a particular triangular brace support qualify as a Pythagorean Triple. You measure the sides of the brace and find them to be 7 inches, 24 inches, and 25 inches. Can you determine if the lengths of the sides of the triangular brace qualify as a Pythagorean Triple?

## Pythagorean Triples

Pythagorean Triples are sets of whole numbers for which the Pythagorean Theorem holds true. The most well-known triple is 3, 4, 5. This means that 3 and 4 are the lengths of the legs and 5 is the hypotenuse. The largest length is always the hypotenuse. If we were to multiply any triple by a constant, this new triple would still represent sides of a right triangle. Therefore, 6, 8, 10 and 15, 20, 25, among countless others, would represent sides of a right triangle.

#### Identifying Pythagorean Triples

Determine if the following lengths are Pythagorean Triples.

1. 7, 24, 25

Plug the given numbers into the Pythagorean Theorem.

\begin{aligned} 7^2+24^2\stackrel{?}{=}25^2 \\ 49+576&=625 \\ 625&=625 \end{aligned}

Yes, 7, 24, 25 is a Pythagorean Triple and sides of a right triangle.

2. 9, 40, 41

Plug the given numbers into the Pythagorean Theorem.

\begin{aligned} 9^2+40^2\stackrel{?}{=}41^2 \\ 81+1600&=1681 \\ 1681&=1681\end{aligned}

Yes, 9, 40, 41 is a Pythagorean Triple and sides of a right triangle.

3. 11, 56, 57

Plug the given numbers into the Pythagorean Theorem.

\begin{aligned} 11^2+56^2&\stackrel{?}{=}57^2 \\ 121+3136&=3249 \\ 3257&\neq 3249 \end{aligned}

No, 11, 56, 57 do not represent the sides of a right triangle.

Example $$\PageIndex{1}$$

Earlier, you were asked to utilize your knowledge of the Pythagorean Theorem to determine if the lengths of a particular triangular brace support qualify as a Pythagorean Triple.

Solution

Since you know that the sides of the brace have lengths of 7, 24, and 25 inches, you can substitute these values in the Pythagorean Theorem. If the Pythagorean Theorem is satisfied, then you know with certainty that these are indeed sides of a triangle with a right angle:

\begin{aligned} 7^2+24^2&\stackrel{?}{=} 25^2 \\ 49+576&=625 \\ 625&=625\end{aligned}

The Pythagorean Theorem is satisfied with these values as a lengths of sides of a right triangle. Since each of the sides is a whole number, this is indeed a set of Pythagorean Triples.

Example $$\PageIndex{2}$$

Determine if the following lengths are Pythagorean Triples.

5, 10, 13

Solution

Plug the given numbers into the Pythagorean Theorem.

\begin{aligned}5^2+10^2 &\stackrel{?}{=} 13^2 \\ 25+100=169 \\ 125\neq 169 \end{aligned}

No, 5, 10, 13 is not a Pythagorean Triple and not the sides of a right triangle.

Example $$\PageIndex{3}$$

Determine if the following lengths are Pythagorean Triples.

8, 15, 17

Solution

Plug the given numbers into the Pythagorean Theorem.

\begin{aligned} 8^2+15^2&\stackrel{?}{=} 17^2 \\ 64+225&=289 \\ 289&=289\end{aligned}

Yes, 8, 15, 17 is a Pythagorean Triple and sides of a right triangle.

Example $$\PageIndex{4}$$

Determine if the following lengths are Pythagorean Triples.

11, 60, 61

Solution

Plug the given numbers into the Pythagorean Theorem.

\begin{aligned} 11^2+60^2&\stackrel{?}{=} 61^2 \\ 121+3600=3721 \\ 3721&=3721 \end{aligned}

Yes, 11, 60, 61 is a Pythagorean Triple and sides of a right triangle.

## Review

1. Determine if the following lengths are Pythagorean Triples: 9, 12, 15.
2. Determine if the following lengths are Pythagorean Triples: 10, 24, 36.
3. Determine if the following lengths are Pythagorean Triples: 4, 6, 8.
4. Determine if the following lengths are Pythagorean Triples: 20, 99, 101.
5. Determine if the following lengths are Pythagorean Triples: 21, 99, 101.
6. Determine if the following lengths are Pythagorean Triples: 65, 72, 97.
7. Determine if the following lengths are Pythagorean Triples: 15, 30, 62.
8. Determine if the following lengths are Pythagorean Triples: 9, 39, 40.
9. Determine if the following lengths are Pythagorean Triples: 48, 55, 73.
10. Determine if the following lengths are Pythagorean Triples: 8, 15, 17.
11. Determine if the following lengths are Pythagorean Triples: 13, 84, 85.
12. Determine if the following lengths are Pythagorean Triples: 15, 16, 24.
13. Explain why it might be useful to know some of the basic Pythagorean Triples.
14. Prove that any multiple of 5, 12, 13 will be a Pythagorean Triple.
15. Prove that any multiple of 3, 4, 5 will be a Pythagorean Triple.

## Vocabulary

Term Definition
Pythagorean Triple A Pythagorean Triple is a set of three whole numbers $$a$$, $$b$$ and $$c$$ that satisfy the Pythagorean Theorem, $$a^2+b^2=c^2$$.