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  • Level: GCSE
  • Subject: Maths
  • Word count: 1172

BEYOND PYTHAGORAS

Extracts from this document...

Introduction

Sohel Patel                                                                                Year 11

BEYOND PYTHAGORAS

        In this report, I am going to investigate the Pythagorean Triple. The Pythagoras Theorem was invented by Pythagoras, a Greek mathematician and philosopher who lived in the 6th centaury BC.

        The Pythagoras Theorem only works in right-angled triangles, where there are three different lengthed sides, one short, one medium, and the other long. A Pythagorean Triple is when a set of numbers satisfy the condition: Shortest side2 + Medium side2 =Longest side2. Also all the sides have to have positive integers.

        Here is an example of a Pythagorean Triple:

        The above triangle is a Pythagorean Triple because it satisfies the condition with all its sides being a positive integer. I will now work out the perimeter and area of the above Pythagorean Triple.

There are also other Pythagorean Triples. Here they are:

        Both the triangles 2) and 3) are Pythagorean Triples because they satisfy the condition and all their sides have a positive integer.

        Here is a table showing the results of the 3 Pythagorean Triples:

Triangle No.

Shortest side

Medium side

Longest side

Perimeter

Area

1)

3

4

5

12

62

2)

5

12

13

30

302

3)

7

24

25

56

842

        From the above table, I can see a few patterns emerging.

...read more.

Middle

        From now on, I am going to abbreviate each quantity and give it a symbol to make it easier for me. Here is a table of the quantities and the symbols that I am going to give them:

Quantity

Symbol

Triangle Number

T

Shortest side

S

Medium side

M

Longest side

L

Perimeter

P

Area

A

        There are two types of sequences, Linear sequences and Quadratic sequences.  A Linear sequence is when the difference between each number is constant, but a Quadratic sequence is when only the second difference between the numbers is constant. S is a Linear sequence:

        However, M and L are Quadratic sequences:

        This is the nthterm formula to find out S:

This is the nth term formula to find out the M:

        As it has been proved earlier that M+1=L, the formula for L will be the same as M plus 1:

Here are the formulas for each side:

        I am now going to generate 3 more Pythagorean Triples using the formulae above.

        Here is a table showing the results of the 8 Pythagorean Triples:

T

S

M

L

P

A

1)

3

4

5

12

62

2)

5

12

13

30

302

3)

7

24

25

56

842

4)

9

40

41

90

1802

5)

11

60

62

132

3302

6)

13

84

85

182

5462

7)

15

112

113

240

8402

8)

17

144

145

306

12242

        I am now going to show by algebraic manipulation, that S2+M2=L2:

...read more.

Conclusion

Now, I will draw a table showing the results of the first 5 Pythagorean Triples.

T

S

M

L

P

A

1)

6

8

10

24

242

2)

10

24

26

60

1202

3)

14

48

50

112

3362

4)

18

80

82

180

7202

5)

22

120

124

266

13202

        As you can see from the above table, the patterns go on and not only that, other patterns have emerged. This is that the difference between M goes up in multiples of 8 each time, which is double to the multiples of 4 (in the difference), it went up by in the previous set.  Also, the Perimeters for these triangles are double to the previous set. I thought this would happen because I doubled all the three sides and so obviously, the Perimeters would double!

        Again, S is a Linear sequence and M and L are Quadratic sequences:

This is the nthterm formula to find out S:

This is the nth term formula to find out the M:

        As it has been proved earlier that M+2=L, the formula for L will be the same as M plus 2:

Here are the formulas for each side:

        From this investigation I have learnt that other Pythagorean Triplet Families can be generated by using a different scale and that there are a lot of other families waiting to be investigated!

...read more.

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