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Pythagorean triplets

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Introduction

Thomas Brown        G.C.S.E Maths Coursework        Mr Tims

11R2        Beyond Pythagoras        Pg

Introduction

We are to investigate the conditions and theory of Pythagorean triplets. Pythagoras’ theorem states: in any right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. For numbers to be Pythagorean triplets they have to satisfy the condition:

a2 + b2= c2image11.pngimage00.pngimage01.png

image27.pngimage35.pngimage20.png

This may be rearranged to give the  a2 =  c2 –  b2    or   b2 = c2 a2, which are useful when calculating one of the shorter sides.

A simple example of this is these numbers: 3 , 4 , 5

                                                        Because 32  =  3 * 3 = 9image42.pngimage47.png

42 = 4 * 4 = 16image53.png

52= 5 * 5 = 25

image03.pngimage13.pngimage04.pngimage02.png

32 + 42 = 9 + 16 = 25 = 52image05.png

This is the 1st Pythagorean Triple

Another example is: 5 , 12 , 13

image07.pngimage06.png

image08.png

image09.png

image02.png

image13.pngimage12.pngimage04.pngimage10.png

image14.png

image15.png

Another Example is: 7 , 24 , 25

image16.png

image17.pngimage02.pngimage18.png

image19.pngimage04.png

image21.pngimage13.png

image22.png

...read more.

Middle

Length of

Shortest side

Length of middle side

Length of longest side

Perimeter

Area

3

4

5

12

6

5

12

13

30

30

7

24

25

56

84

We can now put these results into a table:

image23.png

To  find the lengths of the sides of the 4th and 5th triplets I have to try and recognize a pattern        :

Shortest Side        

3image24.png

          `                      + 2

5image24.png

                              + 2

7

The difference between the numbers of the shortest side seems to be 2. I can now confidently say that the shortest side in the 4th triple will calculate to be 9.

Middle side        

image25.png

4

+   8

12+4

+ 12

24

I have found that the difference (2) of the difference (1) is 4,so,

I can estimate that the length of the middle side for triplet 4 will be 40.

Because       12 + 4 = 16

image02.png

24 + 16 = 40

Longest Side

Here are the lengths of the longest sides:

                Middle                Longestimage26.png

   4                +1             5

                   12                +1            13

                   24                +1              25

The length of the longest side seems to be the length of the middle side +1

Here are my results for the 4th triple:

Length of               Length of        Length of             Perimeter       Areaimage28.pngimage28.pngimage28.pngimage28.png

smallest side     middle side        longest side          units          sq  units

image29.png

        9                40                        41                90                180

I can now see if these numbers satisfy the condition  a2 + b2= c2

image30.png

image32.pngimage31.png

image33.png

This is the 4th Pythagorean triple

I can now put this new result into my table and work out the 5th triple

image34.png

Length of

Shortest side

Length of middle side

Length of longest side

Perimeter

Area

3

4

5

12

6

5

12

13

30

30

7

24

25

56

84

9

40

41

90

180

11

60

61

132

330

...read more.

Conclusion

Longest side = +1 to middle side

Finding a nth term for the sequences:

Smallest Side

n                         1                    2                  3                4                 5image40.png

Smallest side     3                    5                 7                 9                11                 image29.png

image41.pngimage41.pngimage41.pngimage41.png

Difference2                   2                  2                  2

After studying the grid I have found that the formulae is 2n + 1

Examples :  1 (n) * 2 + 1 = 3

                3 (n) * 2 + 1 = 7

                5 (n) * 2 + 1 = 11

Middle Side

n                       1                    2                       3                4                5image43.pngimage44.png

Middle Side               4                    12             24               40             60image41.pngimage41.pngimage41.pngimage41.pngimage41.png

image45.pngimage45.pngimage46.pngimage45.png

Diff 18                   12                6              20

image48.png

Diff 2                                        4                4                 4image49.pngimage50.png

½ 2nd Diff                                        2                2

This sequence is quadratic therefore we know it will include and n2. Now I know the sequences ½ 2nd difference is 2 I Know the first part to my formulae will be 2n2.

 I will now attempt to work out the 2nd part to my formulae by using a table

     Triple

      2n2

          Middle side

Difference between 2n2 and Middle

1

2

4

2image51.png

2

8

12

4image52.png

3

18

24

6

4

32

40

8

5

50

60

10

I can see now that the difference between n2 and middle side is 2 so I can now say that the formulae will be 2 (n2 ) + 2.

Examples -  2 (1(n)2) + 2n = 4

-  2 (3(n)2) + 2n = 24

-  2 (5(n)2) + 2n = 60

...read more.

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