# BEYOND PYTHAGORAS

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Introduction

Sohel Patel Year 11

BEYOND PYTHAGORAS

In this report, I am going to investigate the Pythagorean Triple. The Pythagoras Theorem was invented by Pythagoras, a Greek mathematician and philosopher who lived in the 6th centaury BC.

The Pythagoras Theorem only works in right-angled triangles, where there are three different lengthed sides, one short, one medium, and the other long. A Pythagorean Triple is when a set of numbers satisfy the condition: Shortest side2 + Medium side2 =Longest side2. Also all the sides have to have positive integers.

Here is an example of a Pythagorean Triple:

The above triangle is a Pythagorean Triple because it satisfies the condition with all its sides being a positive integer. I will now work out the perimeter and area of the above Pythagorean Triple.

There are also other Pythagorean Triples. Here they are:

Both the triangles 2) and 3) are Pythagorean Triples because they satisfy the condition and all their sides have a positive integer.

Here is a table showing the results of the 3 Pythagorean Triples:

Triangle No. | Shortest side | Medium side | Longest side | Perimeter | Area |

1) | 3 | 4 | 5 | 12 | 62 |

2) | 5 | 12 | 13 | 30 | 302 |

3) | 7 | 24 | 25 | 56 | 842 |

From the above table, I can see a few patterns emerging.

Middle

From now on, I am going to abbreviate each quantity and give it a symbol to make it easier for me. Here is a table of the quantities and the symbols that I am going to give them:

Quantity | Symbol |

Triangle Number | T |

Shortest side | S |

Medium side | M |

Longest side | L |

Perimeter | P |

Area | A |

There are two types of sequences, Linear sequences and Quadratic sequences. A Linear sequence is when the difference between each number is constant, but a Quadratic sequence is when only the second difference between the numbers is constant. S is a Linear sequence:

However, M and L are Quadratic sequences:

This is the nthterm formula to find out S:

This is the nth term formula to find out the M:

As it has been proved earlier that M+1=L, the formula for L will be the same as M plus 1:

Here are the formulas for each side:

I am now going to generate 3 more Pythagorean Triples using the formulae above.

Here is a table showing the results of the 8 Pythagorean Triples:

T | S | M | L | P | A |

1) | 3 | 4 | 5 | 12 | 62 |

2) | 5 | 12 | 13 | 30 | 302 |

3) | 7 | 24 | 25 | 56 | 842 |

4) | 9 | 40 | 41 | 90 | 1802 |

5) | 11 | 60 | 62 | 132 | 3302 |

6) | 13 | 84 | 85 | 182 | 5462 |

7) | 15 | 112 | 113 | 240 | 8402 |

8) | 17 | 144 | 145 | 306 | 12242 |

I am now going to show by algebraic manipulation, that S2+M2=L2:

Conclusion

Now, I will draw a table showing the results of the first 5 Pythagorean Triples.

T | S | M | L | P | A |

1) | 6 | 8 | 10 | 24 | 242 |

2) | 10 | 24 | 26 | 60 | 1202 |

3) | 14 | 48 | 50 | 112 | 3362 |

4) | 18 | 80 | 82 | 180 | 7202 |

5) | 22 | 120 | 124 | 266 | 13202 |

As you can see from the above table, the patterns go on and not only that, other patterns have emerged. This is that the difference between M goes up in multiples of 8 each time, which is double to the multiples of 4 (in the difference), it went up by in the previous set. Also, the Perimeters for these triangles are double to the previous set. I thought this would happen because I doubled all the three sides and so obviously, the Perimeters would double!

Again, S is a Linear sequence and M and L are Quadratic sequences:

This is the nthterm formula to find out S:

This is the nth term formula to find out the M:

As it has been proved earlier that M+2=L, the formula for L will be the same as M plus 2:

Here are the formulas for each side:

From this investigation I have learnt that other Pythagorean Triplet Families can be generated by using a different scale and that there are a lot of other families waiting to be investigated!

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

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