# Beyond Pythagoras.

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Introduction

Beyond Pythagoras

Having done Pythagoras’ theorem in detail in and out of class, I am familiar with what his theorem states. I can show that his theorem works on the following triples using details and substitution.

## The 3-4-5 triangle

(3) 2 + (4) 2 = (5) 2

=> 9 + 16 = 25 TRUE: 3-4-5 is a Pythagorean triple.

## The 5-12-13 triangle

(5) 2 + (12) 2 = (13) 2

=> 25 + 122 = 169 TRUE: 5-12-13 is a Pythagorean triple.

The 7-24-25 triangle

(7) 2 + (24) 2 = (25) 2

=> 49 + 576 = 625 TRUE: 7-24-25 is Pythagorean triple.

During the methods, I have observed the following about the numbers and triangles used so far:

- The longest side (hypotenuse) is related to the middle length side. i.e. middle side + 1
- The shortest length side is always an odd number follow the pattern 3-5-7…
- The middle length side is even and a multiple of 4.

I have also been asked to find the area and perimeter of the triangles done in the last exercise. The formula for finding the area of any triangle is ½base X height or base X height. 2 2

## Area and perimeter of the triangle 3-4-5

## Area = ½ * 3 * 4 Perimeter = 3 + 4 + 5

= 6 units2 = 12 units

Middle

Difference

24

+ 16

40

+20

60

+24

84

Finally the side c (The longest length) always seems to be ‘+1’ onto length b giving 41,61,41 triple. I will start at looking at the 9-40-41 triple. This can be seen in the table above.

## The 9-40-41 triangle

(9) 2 + (40) 2 = (41) 2

=> 81 + 1600 = 1681

TRUE: 9-40-41 is a Pythagorean triple.

Following the last example, it can be seen that the theorem still holds. So far, any right angled triangle. a, b, c a2 + b2 = c2

## As suggested, I shall now investigate further right angled triangles where the shortest side (a) is an odd number. Having looked at the short sides (side a) 3,5,7 and 9, I will now continue with 11,13, and 15. The length of the middle side (side b) will continue 60,84 and 112 respectively. The hypotenuse (side c) is side b + 1. For each triangle their Area and perimeter can be seen below.

## The 11-60-61 triangle

(11) 2 + (60) 2 = (61) 2

=> 121 + 3600 = 3721

TRUE: 11-60-61 is a Pythagorean triple.

## The 13-84-85 triangle

## The 15-112-113 triangle

5 | 11 | 60 | 61 | 132 | 330 |

6 | 13 | 84 | 85 | 182 | 546 |

7 | 15 | 112 | 113 | 240 | 840 |

8 | 17 | 144 | 145 | 306 | 1224 |

As with all sequences, I could continue with the table above by working with the columns, already mentioned.

Conclusion

- The longest side (hypotenuse) is not related to the middle length side. For the other triangles it followed a pattern, b + 1 = c
- Previously the shortest length side (a) was always an odd number. The 6-8-10 triangle doesn’t follow this pattern.

But the pattern, which I noticed that was the middle length side (b) is always a multiple of 4 still continues. The area, perimeter and details of the triangles can be seen further on.

## The 6-8-10 triangle

(6)2 + (8) 2 = (10) 2

=> 36 + 64 = 100 TRUE: 6-8-10 is a Pythagorean triple.

## The 9-12-15 triangle

(9)2 + (12) 2 = (15) 2

=> 81 + 144 = 225 TRUE: 9-12-15 is a Pythagorean triple.

Length of shortest side | Length of middle side | Length of longest side | Perimeter | Area |

6 | 8 | 10 | 24 | 24 |

9 | 12 | 15 | 36 | 54 |

If I solve for ‘n’,’n’ will be all the numbers where the perimeter and area will be equal to each other. The solution by trial and error can be time consuming therefore another way would be more appropriate. Solution by graph!

## n | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

y | -50 | -12 | 0 | -2 | -6 | 0 | 28 |

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