# Beyond Pythagoras

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Introduction

## Beyond Pythagoras

Introduction:

a² + b² = c²

The numbers 3, 4 and 5 can be the lengths of the sides of a right-angled triangle.

The perimeter = a + b + c The area = a × b ÷ 2

The numbers 5, 12 and 13 can also be the lengths of the sides of a right-angled triangle.

This is also true for 7, 24 and 25.

These numbers are all called Pythagorean triples because they satisfy the condition.

Aim: I am going to investigate the different values of a, b and c for which the formula a² + b² + c² works. I will also investigate the even and odd values for (a) for which the formula works.

Middle

112

113

240

840

8

17

144

145

306

1224

Formulas:

The formula for a is:

## N a

1 3 = 1 ×2 + 1

2 5 = 2× 2 + 1

3 7 = 3 × 2 + 1

4 9 = 4 ×2 + 1

5 11 = 5 × 2 + 1

n × 2 + 1

So from this I can work out that the general rule for A is 2n + 1.

Now I will try and work out a formula for B.

B | 4 | 12 | 24 | 40 |

N | 1 | 2 | 3 | 4 |

N² | 1 | 4 | 9 | 16 |

Difference between b and n | 2 | 4 | 6 | 8 |

So the formula for B is n² + n × 2 + n²

C is one bigger than B so the formula for C is n² + n × 2 + n² + 1.

The formula for P = a + b + c so that is (2n + 1)+(n² + n × 2 + n² )+(n² + n × 2 +n² + 1) = P

For example 7 + 24 + 25 = 56

The formula for area is A × B ÷ 2 so that means (2n + 1)(n² + n ×

Conclusion

B | 8 | 15 | 24 | 35 |

N | 1 | 2 | 3 | 4 |

N² | 1 | 4 | 9 | 16 |

Difference between b and n | 3 | 8 | 15 | 19 |

So the formula for B is n² + 4n + 3

C is two bigger than B so the formula for C is n² + 4n + 3 + 2.

The formula for P = a + b + c so that is (2n + 4) + (n² + 4n + 3 ) + (n² + 4n + 3 + 2) = P .

For example 6 + 8 + 10 = 24

The formula for area is A × B ÷ 2 so that means (2n + 4)(n² + 4n + 3) ÷ 2.

For example 6 × 8 ÷ 2 = 24

Part 3:

I will now change the rules again:

- (a) Must be a multiple of 3
- (b) Must be a multiple of 4
- (c) Must be a multiple of 5

I will also try out my formulas from my previous Pythagorean triples and see if they work.

N | A | B | C | P | Area |

1 | 3 | 4 | 5 | 12 | 6 |

2 | 6 | 8 | 10 | 24 | 24 |

3 | 9 | 12 | 15 | 36 | 54 |

4 | 12 | 16 | 20 | 48 | 96 |

5 | 15 | 20 | 25 | 60 | 150 |

Hi h

This student written piece of work is one of many that can be found in our GCSE Pythagorean Triples section.

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