# Beyond Pythagoras

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Introduction

## Beyond Pythagoras

Introduction.

A Pythagorean triple is a set of three integers a, b and c that specify the lengths of a right angled triangle - that is c2 = a2 + b2. The numbers 3, 4 and 5 is one example. Another way of writing this is (smallest number) 2 + (middle number) 2 = (largest number) 2 An example of the (3,4,5) triangle:

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

In my investigation I will also look at perimeter and area.

The perimeter of this triangle is: 3+4+5=12 units. The area of the triangle is: ½ x 3 x 4= 6 units.

In my investigation I will observe patterns and work out formulae relating a side or measurement to the term number. I will see if each pattern has the same or different formulae. I will start with the shortest side being odd.

1. Shortest side being an odd integer.

Term number (N) | a | b | c | Perimeter | Area |

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

2 | 5 | 12 | 13 | 30 | 30 |

3 | 7 | 24 | 25 | 56 | 84 |

4 | 9 | 40 | 41 | 90 | 180 |

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

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

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

Formulae:

a= 2n+1

This is because there is a difference of two between each number in the pattern. When this happens as a general rule the formulae must have 2n then I found the difference between 2n and the actual length in each case, this happened to be 1.

Example of a=2n+1: The answer to a when n=6 is 13.The answer to the formula: 2x6+1=a

2x6=12

12+1=13 so the formula is correct.

b=2n2 +2n

Middle

a

b

c

Perimeter

Area

1

4

3

5

12

6

2

6

8

10

24

24

3

8

15

17

40

60

4

10

24

26

60

120

5

12

35

37

84

210

Formulae:

a= 2n+2

This is because there is a difference of two between each number in the pattern. When this happens as a general rule the formulae must have 2n then I found the difference between 2n and the actual length in each case, this happened to be 2.

Example of a=2n+2: The answer to a when n=5 is 12.The answer to the formula: 2x5+2=a

2x5=10

10+2=12 so the formula is correct.

b=n2 +2n

This is because there is a second difference between each number. The first difference is not the same every time but the difference between this first difference was a constant of 2. As a general rule you halve this number and put n2 after it, the formula must have n2 in it. Then there is 2n because each time that was the difference between the answer for the first part and the actual answer.

Example of b=n2+2n: The answer to b when n=4 is 24. The answer to the formula:

4^2+2x4=b

4^2=16

2x4=8

16+8=24 So the formula is correct.

c=n2+2n+2

I noticed that the patterns in this length column were exactly the same as in b but they were two numbers more. So I used the same formula for the same reasons as in b but added 2.

Example of c=n2+2n+2: The answer to c when n=3 is 17.

Conclusion

An example of s-L=(0.5s)2+1: The answer to L when s=4 is 5. The formula shows:

(0.5x4)2+1=s to L.

(0.5x4)2=4

4+1=5

So the formula is correct.

5. P=A

In each of the cases I have noticed that there is a clear link between each of the relevant perimeters and areas. In terms of n the area is always half n x perimeter. From this I can derive a way of finding the perimeter when I have the area and the area when I have the perimeter.

Area when I have perimeter:

Formula= A= n/2xP.

This is because in each of the cases there was a clear link between the two. For n=1 the area was always half the perimeter, for n=2 the area was always the same as the perimeter. This continues as so:

n | A |

1 | 0.5*P |

2 | 1*P |

3 | 1.5*P |

4 | 2*P |

5 | 2.5*P |

6 | 3*P |

7 | 3.5*P |

8 | 4*P |

Etc…

Also each of the numbers to times the perimeter by were halve the term number.

Example of A= n/2xP: A perimeter of a 3,4,5 triangle is 12 and the perimeter is 6 also n=1. The formula shows:

A= 1/2x12

A=6 this is correct.

## Perimeter when I have area

Formula= P=2A/n

I found this by making P the subject of the equation from above. By multiplying both sides by two and dividing by n I got this formula.

### Example of P=2A/n: The area of an 8,15,17 triangle is 60, the term number is three and the perimeter is 40. The formula shows:

P=2x60/3

2x60=120

120/3=40 so the formula is correct.

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

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