# Beyond Pythagoras

Extracts from this document...

Introduction

Contents:

###### Introduction 3,4

###### Satisfying the condition/Table 1 5,6

Calculations to find results in sequence 7

Finding the nth term for Table 1 7,8

Perimeter and area for Table 1 9,10,11

Enlarging Triangles 11,12

Conclusion 13

Table of notations:

S = Smallest side

M = Middle side

L = Longest side

A = Area of triangle

P = Perimeter of triangle

E = Enlargement

N = Position in table (used in nth term)

Introduction:

The Greek mathematician and philosopher Pythagoras developed Pythagoras’ Theorem. Several philosophers who no doubt had a considerable influence on his future life had taught him from an early age.

Pythagoras’ Theorem:

C

A B

Pythagoras’ Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides

i.e. BC2 = AB2 + AC2

(Refers to the diagram above)

In a triangle such as:

5

3

4

32 + 42 = 52

The sum of the lengths of the two short sides equals the length of the hypotenuse.

Middle

60

61

132

330

6

13

84

85

182

546

7

15

112

113

240

840

8

17

144

145

306

1224

9

19

180

181

380

1710

10

21

220

221

462

2310

Calculations to find out the next results in the sequence:

I discovered that for the shortest side the difference was always +2 (3,5,7,9….). For the middle and longest side the difference was always the same, and followed the pattern of the 4 times tables (e.g. 8,12,16,20…).

Finding the nth term for each column in Table 1:

N | 1 | 2 | 3 | 4 | 5 | 6 | |||||

Short side | 3 | 5 | 7 | 9 | 11 | 13 | |||||

1st difference | 2 | 2 | 2 | 2 | 2 |

Column A

Trial and error was my method to find the nth term.

### When n=1

2n = 2 x (1) = 2

2n + 1 = 2 x (1) + 1

2 + 1 = 3

### When n=2

2n + 1= 2(2) +1

= 4 + 1

= 5

This formula works, therefore for this column:

2n + 1 is the correct formula

For column B, the second differences were the same, so I knew it was a squared formula.

## N | 1 | 2 | 3 | 4 | |||

Middle side | 4 | 12 | 24 | 40 | |||

1st Difference | 8 | 12 | 16 | ||||

2nd Difference | 4 | 4 |

Column B

I then used trial and error

### When n=1

2(12) + 2(1)

2 + 2 = 4

### When n=2

2(32) + 2(3)

18 + 6 = 24

This formula works, therefore for this column:

2n2 + 2n is the correct formula

Conclusion

S = E (2n+1) M = (2n2+2n) L = E (2n2+2n+1)

For the area, I knew that the units are UNITS SQUARED, so….

If the ratio of lengths is 1:E

1:E2

Then Ratio of areas is 1:E2

Therefore, this means that for the area column:

A = E2 n (n+1) (2n+1)

###### Perimeter is a length so he ratio of perimeter is 1:E

P = E (4n2 + 6n + 2)

Conclusion:

In this project I have investigated several possible Pythagorean triples, and by trial and error found formulas to express this.

I have found that for any whole number ‘n’ a Pythagorean triple is given by:

## S = 2n + 1

M = 2n2 + 2n

L = 2n2 + 2n + 1

Further triples can be found by enlarging these by ‘E’.

## S = E (2n + 1)

M = E (2n2 + 2n)

L = E (2n2 + 2n + 1)

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

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