Three positive numbers a, b, and c make up a Pythagorean triple if their sum, a^{2} + b^{2}, equals their sum, c^{2}. A standard way to write such a triple is (a, b, c). One of the most common well-known example of Pythagorean triples is (3, 4, 5).

**Complete List of 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.

Some most commonly used triplets are:

a | b | c |
---|---|---|

3 | 4 | 5 |

5 | 12 | 13 |

6 | 8 | 10 |

7 | 24 | 25 |

8 | 15 | 17 |

9 | 40 | 41 |

10 | 24 | 26 |

11 | 60 | 61 |

12 | 35 | 37 |

13 | 84 | 85 |

15 | 112 | 113 |

16 | 63 | 65 |

18 | 24 | 30 |

17 | 144 | 145 |

19 | 180 | 181 |

20 | 21 | 29 |

20 | 99 | 101 |

21 | 220 | 221 |

28 | 45 | 53 |

36 | 77 | 85 |

39 | 80 | 89 |

65 | 72 | 97 |

**Pythagoras Theorem Examples and Questions**

**Example 1: ∆ ABC is right-angled at C. If AC = 5 cm and BC = 12 cm find the length of AB.**

Refer to the figure given on the right.

As the triangle is right-angled, by Pythagoras theorem,

AB² = AC² + BC²

AB² = 5² + 12²

AB² = 25 + 144

AB² = 169 = 13²

Hence, AB = 13 cm

**Example 2 : The side of a square is given to be 4 cm. Find the length of the diagonal of a square.**

Solution: We need to find the length of the diagonal AC. As we know, all sides of a square are equal and each angle is 90 degrees, ADC is a right-angled triangle.

By Pythagoras theoram,

AD² + CD² = AC²

AC² = 4² + 4² = 16 + 16

AC² = 32, So, AC= 4√2

The diagonal of the square is 4√2 cm.