In these lessons, we will learn

- Pythagorean Triples
- Families of Pythagorean Triples
- Pythagorean Triples and Right Triangles
- Solving Problems using the Pythagorean Triples
- How to generate Pythagorean Triples

Pythagorean triples are formed by positive integers *a*, *b* and *c*, such that *a*^{2} + *b*^{2} = *c*^{2}. We may write the triple as (*a*, *b*, *c*)

For example, the numbers 3, 4 and 5 form a Pythagorean Triple because 3^{2} + 4^{2} = 5^{2}. There are infinitely many Pythagorean triples.

( 3 , 4 , 5 ) | ( 5, 12, 13) | ( 7, 24, 25) | ( 8, 15, 17) |

( 9, 40, 41) | (11, 60, 61) | (12, 35, 37) | (13, 84, 85) |

(16, 63, 65) | (20, 21, 29) | (28, 45, 53) | (33, 56, 65) |

(36, 77, 85) | (39, 80, 89) | (48, 55, 73) | (65, 72, 97) |

If we multiply each number of a Pythagorean triple by the same number, we form another Pythagorean triple.

For example, (6, 8, 10) is a family of the Pythagorean triple (3, 4, 5) because it can be obtained by 2 × 3 = 6, 2 × 4 = 8, 2 × 5 = 10. If we multiply (3, 4, 5) by 3, we get another triple (9, 12, 15). This can be repeated with different multiples.

Similarly, the family for (5, 12, 13) would be (10, 24, 26), (15, 36, 39) and so on.

When we make a triangle with sides whose lengths are the Pythagorean Triples, the triangle will form a right triangle. This follows from the converse of the Pythagorean Theorem.

Memorizing some common Pythagorean triples can helpful.

For example, if we see that the sides of a triangle form a Pythagorean triple then we know that it is a right triangle.

Pythagorean triples may also help us to find the missing side of a right triangle faster. If two sides of a right triangle form part of a triple then we can know the value of the third side without having to calculate using the Pythagorean theorem.

*Example:*

Find the value of *x *

**Solution:**

Check for Pythagorean triple:

Get the ratio of the two given sides:

12 : 20 = 3 : 5 ( divide by 4 )

From the ratio, we know that it is a Pythagorean triple.

So, *x* = 4 × 4 = 16 cm

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.