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  • Level: GCSE
  • Subject: Maths
  • Word count: 2424
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    Beyond Pythagoras P.1 Pythagoras Theorem is a2+b2= c2 'a' is being the shortest side, 'b' being the middle side and 'c' being

    Extracts from this document...

    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

    image00.png

    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

    image01.png

    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

    image02.png

    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.

    ...read more.

    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

    ...read more.

    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.

    ...read more.

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