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  • Level: GCSE
  • Subject: Maths
  • Word count: 1607

Beyond Pythagoras.

Extracts from this document...

Introduction

Mathematics Coursework - Beyond Pythagoras

Sam Coates

In a right angled triangle, Pythagoras came to the conclusion that on a right angled triangle the addition of the squares of the two smallest sides are equal to the square of the longest square. Thus the equation - a²+b²=c²

Pythagorean Triples are numbers which are positive integers that comply with the rule.

For example, the numbers 3, 4, and 5 satisfy the condition

3² + 4² = 5²

because 3² = 3x3 =9

4² = 4x4 = 16

5² = 5x5 = 25

and so      

3² + 4² = 9 + 16 = 25 = 5²

Research

Testing the Theory

I will now have to find out if the following sets of numbers satisfy a similar condition of (smallest number) ² + (middle number) ² = (largest number) ².

a) 5, 12, 13image00.png

5² + 12² = 25 + 144 = 169 = 13²image02.pngimage02.pngimage03.pngimage01.png

b) 7, 24, 25image04.png

7² + 24² = 49 + 576 = 625 = 25²image05.pngimage01.png

image06.png

The right angled triangle satisfies the condition.

Term

Shortest Side

Middle Side

Longest Side

Perimeter

Area

1

3

4

5

12

6

2

5

12

13

30

30

3

7

24

25

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

2310

2) Perimeter


I looked at the table and noticed that there was only 1 difference between the length of the middle side and the length of the longest side. And also if you can see in the shortest side column, the numbers increase by the odd numbers from 3 upwards. I have also noticed that the area is

½ (shortest side) x (middle side).

...read more.

Middle

2

5

2x2=4

4+1=5 (correct)

From looking at my table of results and carrying out some testing of rules, I noticed that ‘an + n  = b’. So I took my formula for ‘a’ (2n + 1) multiplied it by ‘n’ to get ‘2n2 + n’. I then added my other ‘n’ to get ‘2n2 + 2n’. This is a parabola as you can see from the equation and also the graph

 I will now test it using the first three terms.

2 x 1² + 2 x 1 = 4

2 x 1 + 2 = 4

2 + 2 = 4

4 = 4

My formula works for the first term; so, I will now check it in the next term.

2 x 2² + 2 x 2 = 12

2 x 4 + 4 = 12

8 + 4 = 12

12 = 12

My formula works for the 2nd term. If it works for the 3rd term I can safely say that

2n² + 2n is the correct formula.

2 x 3² + 2 x 3 = 24

2 x 9 + 6 = 24

18 + 6 = 24

24 = 24

My formula also works for the 3rd term. I am now certain that 2n² + 2n is the correct formula for finding the middle side.

Middle side = 2n² + 2n

I now have the much easier task of finding a formula for the longest side.

...read more.

Conclusion

1" rowspan="10">

+1

12

8

24

12

4

40

16

4

60

20

4

84

24

4

112

28

4

144

32

4

180

36

4

220

40

4

2nd Difference is four

Difference between shortest and middle length sides generalisations.

A

Difference

B

3

1

4

5

7

12

7

17

24

9

31

40

11

49

60

13

71

84

15

97

112

17

127

144

19

161

180

21

199

220

1  7  17  31  49  71  97  127  161  199

  6 10  14  18  22  26  30   34    38

2nd Difference is four

Perimeter

1st Difference

2nd Difference

3rd Difference

2nd Difference

1st Difference

Area

12

6

30

18

24

30

56

26

8

30

54

84

90

34

8

12

42

96

180

132

42

8

12

54

150

330

182

50

8

12

66

216

546

240

58

8

12

78

294

840

306

66

8

12

90

384

1224

380

74

8

12

102

486

1710

462

82

8

12

114

600

2310

...read more.

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