# 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:

The numbers 5, 12 and 13 satisfy the connection.

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.

7² + 24² = 25²

49 + 576 = 625

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.

Middle

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.

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

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

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