# Beyond Pythagoras P.1 Pythagoras Theorem is a2+b2= c2 'a' is being the shortest side, 'b' being the middle side and 'c' being

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Introduction

Mahmoud ElsherifBeyond PythagorasP.1

Pythagoras Theorem is a2+b2= c2 ‘a’ is being the shortest side, ‘b’ being the middle side and ‘c’ being the longest side (hypotenuse) of a right angled triangle.

The numbers 3,4,5 satisfy this condition and so

32+ 42=52

Because 32= 3*3=9

42=4*4=16

52=5*5=25

32+ 42=52

9+16=25

25=25

This proves Pythagoras Theorem goes with the right angled triangle with the numbers 3,4,5. Next I shall prove that Pythagoras’s Theorem applies to 5,12,13 right angled triangle.

52+122=132

Because 52= 5*5=25

122= 12*12=144

132= 13*13=169

Mahmoud ElsherifBeyond PythagorasP.2

This satisfies the Theorem of Pythagoras’s goes with these numbers 5,12,13. Finally I shall prove that Pythagoras’s Theorem applies to 7,24,25 right angled triangle.

72+ 242=252

Because 72= 7*7=49

242= 24*24= 576

252=25*25=625.

So

a2+b2=c2

72+242=252

49+576=625

This proves Pythagoras Theorem goes with the right angle triangle with the sides 7,24,25

Shortest Side | Middle Side | Longest Side |

3 | 4 | 5 |

5 | 12 | 13 |

7 | 24 | 25 |

9 | 40 | 41 |

11 | 60 | 61 |

13 | 84 | 85 |

Mahmoud ElsherifBeyond PythagorasP.3

I shall find the prediction of the shortest side first.

3,5,7

It goes up in 2 so in my conclusion so it will become

3,5,7,9,11,13

Now I will find the difference between them.

The difference is 2

Next I shall find the prediction of the middle side next.

4,12,24

It goes up by 4,8,12. So in my conclusion I think it will become 4,8,12,16,20,24

So it will be 4,12,24, 40, 60 84.

The difference is 4,8,12. Now I shall find the difference and it is n*4

Next I shall find the prediction of the longest side next.

5,13,25

It goes up by 4,8,12.

Middle

Middle

2n2+2n + 2n2+2n

4n4+4n3

4n3+ 4n2

4n2 +8n3+4n4

Longest

2n2+2n+1 + 2n2+2n+1

4n4+4n3+2n2

4n3+ 4n2+2n

2n2+2n+1

4n4+8n3+8n2+4n+1

(4n2 +8n3+4n4)= (4n4+8n3+8n2+4n+1)- (4n+ 4n2+1)

Finally I will investigate the shortest Term.

Mahmoud ElsherifBeyond PythagorasP.7

Shortest Term2= Longest Term2- Middle Term2

(nth term)2= (nth term)2 + (nth term)2

(2n2+1)2 =(2n2+2n+1) 2 - (2n+2n2)2

Shortest

2n+1+ 2n+1

4n2+ 2n

2n + 1

4n+ 4n2+1

Middle

2n2+2n + 2n2+2n

4n4+4n3

4n3+ 4n2

4n2 +8n3+4n4

Longest

2n2+2n+1 + 2n2+2n+1

4n4+4n3+2n2

4n3+ 4n2+2n

2n2+2n+1

4n4+8n3+8n2+4n+1

(4n+ 4n2+1)= (4n4+8n3+8n2+4n+1)-(4n2 +8n3+4n4)

Now I have finished that I will start having even numbers, to see if Pythagoras’s Theorem works. I shall do this all over again but with an even short side.

Shortest Side | Middle Side | Longest Side |

6 | 8 | 10 |

10 | 24 | 26 |

14 | 48 | 50 |

18 | 80 | 82 |

22 | 120 | 122 |

26 | 168 | 170 |

Mahmoud Elsherif Beyond Pythagoras P.8

I shall find the prediction of the shortest side first.

6,10,14

It goes up in 4 so in my conclusion so it will become

6,10,14,18,22,26.

Now I will find the difference between them.

The difference is 4

Next I shall find the prediction of the middle side next.

8,24,48

It goes up by 8,16,24. So in my conclusion I think it will become 8,16,24,32,40,48

Conclusion

Shortest Side | Middle Side | Longest Side |

6 | 8 | 10 |

10 | 24 | 26 |

14 | 48 | 50 |

18 | 80 | 82 |

22 | 120 | 122 |

26 | 168 | 170 |

Mahmoud ElsherifBeyond PythagorasP.14

To get the third part you have to times the first part by three

9 | 12 | 15 |

15 | 36 | 39 |

21 | 72 | 75 |

27 | 120 | 123 |

33 | 180 | 183 |

39 | 252 | 255 |

Finally to get the fourth part you have to times the first part by four.

Shortest Side | Middle Side | Longest Side |

12 | 16 | 20 |

20 | 48 | 52 |

28 | 96 | 100 |

36 | 160 | 164 |

44 | 240 | 244 |

52 | 336 | 340 |

So my theory is correct with the times.

For the shortest formula it is x (2n+1)

For the middle formula it is x (2n2+2n)

For the longest formula it is x (2n2+2n+1)

My formula is right

A= B +(B+ X)

You get for the top part of every table you divide it by two the B that is. B/2

Same with the first except in the second you divide it by three. B/3 For 5,12,13.

Then if it’s the next slot then you have to plus the number divide by plus one from the previous one to get next one. For 7,24,25 divide 24/4 +x.

That is my conclusion.

So my theory is correct with B. I can prove it by this

Mahmoud ElsherifBeyond PythagorasP.15

13= 12+ (12+1) divide b/3 in the brackets

132=122+52

25= 24+ (24+2) divide b/3 in the brackets

252=242+102

25= 24+ (24+1) divide b/4 in the brackets

252=242+72

50= 48+ (48+2) divide b/4 in the brackets

502=482+142

I have gotten all the evidence right here to prove my theory.

Both of these methods I have used the table and the dividing b have also proven the theory and gave me all the evidence I needed to finish this.

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

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