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Beyond Pythagoras

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Introduction

BEYOND PYTHAGORAS

By: Megan Garibian

10A

What this coursework has asked me to do is to investigate and find a generalisation, for a family of Pythagorean triples. This will include odd numbers and even numbers.

I am going to investigate a family of right-angled triangles for which all the lengths are positive integers and the shortest is an odd number.

I am going to check that the Pythagorean triples (5,12,13) and (7,24,25) cases work; and then spot a connection between the middle and longest sides.

The first case of a Pythagorean triple I will look at is:

image00.pngimage01.png

The numbers 5, 12 and 13 satisfy the connection.image04.pngimage03.pngimage02.png

5² + 12² = 13²

25 + 144 = 169

169 = 13

The second case of a Pythagorean triple I will look at is:

The numbers 7, 24 and 25 satisfy the connection.image01.pngimage00.png

7² + 24² = 25²

49 + 576 = 625image03.pngimage04.pngimage02.png

625 = 25

There is a connection between the middle and longest side. This is that there is a one number difference.

So if M= middle and L= longest

L = M + 1

I am going to use the triples, (3,4,5), (5,12,13) and (7,24,25) to find other triples. Then I will put my results in a table and look for a pattern that will occur.

...read more.

Middle

image05.pngimage06.pngimage05.pngimage06.png

image07.pngimage06.png

         2n +1                            Sxn +n                        M+1

I have established a connection between n and S, S and M, and, M and L.

I can see if I multiply n by 2 and add 1 to it I get S.

I can see if I multiply S by n and add n to it I get M.

I can see if I add 1 to M I get L.

In another way S, M, and L are forming a sequence. I know how to find the n   term for a sequence so I applied this knowledge and came up with a formula.

By: Megan Garibian

Prediction:

 I am going to use the formula I found to predict the next set of results.

S=  2n +1                M=  Sxn +n                L=  M+1

      2x4 +1                       9x4 +4                     40+1

      8 +1                       36 +4                L = 41                    

S = 9                      M = 40

This means my prediction is, that the next triple will be:

Shortest side = 9

Middle    side = 40

Longest  side = 41

I will prove my prediction by using Pythagoras thermo.

9²+40²=41²

81 + 1600 = 1681           1681 = 41

My prediction was correct. That means my formula works.

S

M

L

1

3

4

5

2

5

12

13

3

7

24

25

4

9

40

41

5

11

60

61

6

13

84

85

7

15

112

113

8

17

144

145

9

19

180

181

10

21

220

221

Now I am going to try my formula a few more times, and also to check that the new data in the table is correct.

...read more.

Conclusion

If n = 50

S = 2n +1                M = 2n²+2n                        L = 2n²+2n +1

   = 2x50 +1                    = 2x50² + (2x50)            = 2x50² + (2x50) +1

   = 100 +1                    = 5000 +         52                    = 5000 + 52 +1

   = 101                    = 5052                           = 5053

To see if the triple will work I am going to use Pythagoras thermo.

101²+5052²=5053²

10201 + 25522704 = 25532809                25532809 = 5053

If n = 19

S = 2n +1                M = 2n²+2n                        L = 2n²+2n +1

   = 2x19 +1                    = 2x19² + (2x19)            = 2x19² + (2x19) +1

   = 38 +1                    = 722 + 21                            = 722 + 21 +1

   = 39                               = 743                           = 744

To see f the triple will work I am going to use Pythagoras thermo.

39²+743²=744²

1521 + 552049 = 553536                553536 = 744

By: Megan Garibian

I have finished investigating this family of Pythagorean triples where the shortest side is an odd number and all three sides are positive integers.

I have checked through cases of Pythagorean triples to see if they satisfy the conditions and spotted a connection between the middle and longest side.

Then I used the first three triples in the sequence to find a pattern and to predict the next results.

I also extended the Pythagorean triples to a sequence of 10, found connections between the short, middle and longest sides. I expressed these connections in algebra and gave a reason for it.

I then found a general rule for the shortest, middle and longest sides in algebra, and proved it was correct by using Pythagoras thermo.

Finally I found more triples with the shortest side being an odd number and all the three sides are positive integers.

By: Megan Garibian

...read more.

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

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