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Introduction

Beyond Pythagoras

(With Odd Number Pythagorean Triples and Even Number Pythagorean Triples For Extension)

By Sheryar Majid

G.C.S.E

Maths Coursework

Winter ‘99

By Sheryar Majid

Introduction

I have been given 3 tasks in this piece of work. These tasks are to help me find out/work out formulae for the patterns between the short, medium and long sides of Pythagorean triples. The formulae I work out will be based on this:

If I have the short side can I work out the middle and long side?

And

If I have the middle side can I work out the short and long side?

And

If I have the long side can I work out the short and middle side?

From these 3 things I have to try and work out the relationships between the 3 sides and the area and perimeter of the Pythagorean triangle.

An example of the 3 sides and the area and perimeter is this:

 Number of triangle (nth term) Length of Shortest Side Length of MiddleSide Length of Longest Side Perimeter Area 1 3 4 5 12 6 2 5 12 13 30 30 3 7 24 25 56 84

From this graph we can already see that the difference between middle side and long side is +1. Also you can see all triples satisfy the condition A2 + B2 = C2, this is because if it were not that than it would not satisfy the condition. I will talk about satisfying the condition later on but basically Pythagorean triples have to satisfy a

condition to work. In the following pages I will do the 3 subtasks given to me and from this I will work out the formulae’s. Everything I do concerning findings I will first predict, then test and then prove.

This is the first of the 3 tasks. In this first task I have been given a set of data to analyze.

Middle

56

84

4

9

40

41

90

180

5

11

60

61

132

330

6

13

84

85

182

546

7

15

112

113

240

840

8

17

144

145

306

1224

9

19

180

181

380

1710

10

21

220

221

462

2318

So you can see that all my patterns worked, except the one I suspected, which was the area one. We can prove that they were correct, by checking if all the Pythagorean triples satisfy the condition. By looking at the table and checking with your calculator that all the triples do satisfy the condition and therefore are correct. Now about the area, the pattern was not right because I had to extend it further to make it right:

 Area 6      + 24    + 30  + 12 30   + 54     + 42  + 12 84   + 96     + 54  + 12 180 + 150  +66   + 12

So we can see that we have to extend to 3 levels to find a pattern and that will now help us to find a formulae.

So we have now seen what is a Pythagorean triple and why it is; because they satisfy the condition. Now my next task is to use this data to help me look for formulae.

I have to now find generalizations about the area, perimeters and sides of the Pythagorean triples beyond those in tasks 1 + 2. I have to look at the lengths of the 3 sides and area and perimeter of the triangles. To do this I have to look at the data I got in the last two tasks and generalize it but look beyond it and at things not obvious in the data. Ultimately from all this I have to work out formulae for the relationship of sides to perimeters and area.

To do this I have to first work out 3 crucial things, which I mentioned in the introduction from which I can work out many other things. These are:

If I have the short side can I work out the middle and long side?

And

If I have the middle side can I work out the short and long side?

And

Conclusion

Extension with Even Numbered Pythagorean Triples

Introduction

After doing my coursework and finding that all my formulae and patterns only worked for odd numbered Pythagorean triples, I thought would extend my work and also do some work on even numbered Pythagorean triples. I went about doing this work the same way as my other work and I first drew out a table similar to the one given to me in my previous work and tried to look for patterns in it like before. This is the table I used:

 Number of triangle (nth term) Length of Shortest Side Length of MiddleSide Length of Longest Side Perimeter Area 1 6 + 2 17.5 +14 18.5 + 14 42 + 30 52.5 + 73.5 2 8 + 2 31.5+ 18 32.5 + 18 72 + 38 126 +121.5 3 10 + 2 49.5+ 22 50.5 + 22 110 + 46 247.5+ 169 4 ? ? ? ? ?

Again I was getting similar patterns between sides of one triple to another and other patterns were similar except the one for area which was a little doubtful again and I had to go to more levels of patterns to get a exact pattern.

But when I had this pattern I could use it to build a bigger graph an test my formulae on. I also thought that if the patterns were similar then the formulae that I had before would be similar and if that was the case my whole investigation would work for both evens and odds.

This is the table I made to test my predictions.

 Number of triangle (nth term) Length of Shortest Side Length of Middle Side Length of Longest Side Perimeter Area 1 6 17.5 18.5 42 52.5 2 8 31.5 32.5 72 126 3 10 49.5 50.5 110 247.5 4 12 71.5 72.5 156 429 5 14 97.5 98.5 210 682.5 6 16 127.5 128.5 272 1020 7 18 161.5 162.5 342 1453.5 8 20 199.5 200.5 420 1995 9 22 241.5 242.5 506 2656.5 10 24 287.5 288.5 600 3450

My predictions were right and with checks with my calculator and also this table, I have found out that all my formulae work for both ODD AND EVEN PYTHGOREAN TRIPLES. The explanation for this could be that the formulae I got was not number dependant (i.e. it did not matter to the formula if it was tried by even or odd numbers).

By Sheryar Majid

Beyond

A2 + B2 = C2

In This Work S, M, L Stand For Short, Middle And Long Side

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