• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  • Level: GCSE
  • Subject: Maths
  • Word count: 1741

Beyond Pythagoras

Extracts from this document...

Introduction

William Kerslake        Mathematics GCSE Coursework

Beyond Pythagoras

Aim

I am conducting this investigation to discover formulae that will allow me to calculate many Pythagorean triples. I will first find formulae for odd Pythagorean triples and then even ones. For each, I will find formulae for ‘a’, ‘b’, ‘c’, perimeter and area, all in terms of ‘n’. After this I hope to discover a general formula for all Pythagorean triples, although it is unlikely that I will.

Contents

Page Number

1

Aim and Contents

2             Odd and Even

Tables of Pythagorean triples

3             Odd

Formula for ‘a’ in terms of ‘n’

4             Odd

Formula for ‘b’ and ‘c’ in terms of ‘a’

5             Odd

Formula for ‘b’ and ‘c’ in terms of ‘n’

6             Odd

Perimeter and Area in terms of ‘n’

7             Even

Formula for ‘a’ in terms of ‘n’

8             Even

Formula for ‘b’ and ‘c’ in terms of ‘a’

9             Even

Formula for ‘b’ and ‘c’ in terms of ‘n’

10           Even

Perimeter and Area in terms of ‘n’

11           Odd and Even

Summary – All Formulae


Odd Pythagorean Triples

Table of first 10 odd triples

n

a

b

c

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

...read more.

Middle

Difference

1

1

3

2

1

2

5

2

1

3

7

2

1

4

9

2

The differences between the numbers show that ‘a’ is changing twice as fast as ‘n’. Therefore, 2n must be part of the formula.

Difference

2n

Difference between n and a

a

Difference

2

2

1

3

2

2

4

1

5

2

2

6

1

7

2

2

8

1

9

2

The differences show that ‘a’ and ‘n’ are changing at the same rate – increasing by 2 each time, thus proving that the formula includes 2n. The difference between ‘a’ and ‘n’ is always 1; therefore, our formula needs to have a +1 included.

From this, I know the formula for ‘a’, in terms of ‘n’, to be;

a  =  2n + 1

I personally cannot see an obvious link between ‘n’ and ‘b’ or ‘c’, so I will use the known formula, a² +b² = c² to discover ‘b’ and ‘c’ in terms of ‘n’.

3² + 4² = 5² and 5² + 12² = 13² both have a hypotenuse one greater than one side (in this case known as ‘b’). Looking at my table of triples I can see that there are more odd triples that follow this pattern. Using this, I should be able to discover ‘b’ and ‘c’ in terms of ‘a’.

3² + 4²        = 5²

a² + b²        = c²

a² + b²        = (b + 1)²

a² + b²        = b² + 1 + 2b

a²         = 2b + 1

2b        = (a² - 1)

b         = (a² - 1)

                2

Remove extra variable ‘c’ from formula

Multiply out brackets

Simplify formula

Rearrange formula

Rearrange formula

From this, I know the formula for ‘b’, in terms of ‘a’, to be;

b = (a² - 1)

     2

From this formula for ‘b’ in terms of ‘a’ I can discover ‘c’ in terms of ‘a’;

a

b

(a² - 1)/2

a²-1

a²+1

(a² + 1)/2

c

3

4

4

8

9

10

5

5

5

12

12

24

25

26

13

13

7

24

24

48

49

50

25

25

9

40

40

80

81

82

41

41

...read more.

Conclusion

((2n + 4) ² - 4)

                        4

b         = 4n² + 8n + 8n +16 - 4

 4

b         = 4n² + 16n + 12

                4

b        = n² + 4n + 3

Insert a = 2n + 4

Multiply out brackets

Sort like numbers/letters

Cancel down to simplest form

And the same with ‘c’;

c         = (a² + 4)

                4

c         = ((2n + 4) ² + 4)

                        4

c         = 4n² + 8n + 8n +16 + 4

 4

c         = 4n² + 16n + 20

                4

c        = n² + 4n + 5

Insert a = 2n + 4

Multiply out brackets

Sort like numbers/letters

Cancel down to simplest form

b = 2n² + 2n

c = 2n² + 2n + 1


Now that I have formulae for ‘a’, ‘b’ and ‘c’ in terms of ‘n’; I can put these into my simple area and perimeter formulae and will then be able to calculate both area and perimeter in terms of ‘n’.

Perimeter

Perimeter = a + b + c

P = (2n + 4) + (n² + 4n +3) + (n² + 4n + 5)

P = 2n + 4 + n² + 4n +3 + n² + 4n + 5

P = 10n + 12 + 2n²

Perimeter = 2n² + 10n + 12

Area

Area = a x b

           2

A = (2n+4) x (n² + 4n +3)

                2

A = 2n³ + 8n² + 6n + 12 + 4n² + 16n

                2

A = 2n³ + 12n² + 22n + 12

        2

A = n³ + 6n² + 11n + 6

Area = n³ + 6n² + 11n + 6


Formulae Summary

These are all the formulae I have discovered in my investigation. I can now workout any odd or even Pythagorean triple and its perimeter and area knowing only the number, n. For example, the 26 odd triple is: a = 53, b = 1404, c = 1405, perimeter = 2862. area = 37206

Odd Pythagorean Triples

a                =        2n + 1

b                =        2n² + 2n

c                =        2n² + 2n + 1

Perimeter        =        4n² + 6n + 2

Area                =        2n³ + 3n² + n

Even Pythagorean Triples

a                =        2n + 4

b                =        n² + 4n + 3

c                =        n² + 4n + 5

Perimeter        =        2n² + 10n + 12

Area                =        n³ + 6n² + 11n + 6

/

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Pythagorean Triples essays

  1. Investigating families of Pythagorean triples.

    The following table shows the values: n a b c a2 b2 c2 1 8 15 17 64 225 289 2 12 35 37 144 1225 1369 3 16 63 65 256 3969 4225 4 20 99 101 400 9801 10201 5 24 143 145 576 20449 21025 6 28

  2. Maths Investigation: Number of Sides

    I will now check if it works using the 2nd term. 4 x 22 - 4(2-1)2 = 12 4 x 4 - 4 x 12 = 12 16 - 4 x 1 = 12 16 - 4 = 12 12 = 12 My formula also works for the 2nd term.

  1. Beyond Pythagoras

    137 9384 9385 69 139 9660 9661 70 141 9940 9941 I now have several different triangles, which I think is easily enough to find a relationship between each side. An easier and faster way to work out the sides would be by using the nth term.

  2. Investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean ...

    between 4n�gets larger and larger, the thing you notice is that the difference in the 2nd term between 4n�and the middle side is the middle side for the term before. This goes for all the other terms from the 2nd.

  1. Beyond Pythagoras.

    I will try multiplying the squared numbers by 2. 2 x 1? = 2 2 x 2? = 8 2 x 3? =18 I seem to be getting somewhere with this formula. I think that the first part of it will be 2n?.

  2. For this piece of work I am investigating Pythagoras. Pythagoras was a Greek mathematician.

    I will try multiplying the squared numbers by 2. 2 x 1� = 2 2 x 2� = 8 2 x 3� =18 I seem to be getting somewhere with this formula. I think that the first part of it will be 2n�.

  1. Beyond Pythagoras

    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.

  2. Maths GCSE coursework: Beyond Pythagoras

    So we must look at 2n Sequence 3 5 7 9 11 13 Value of 2n 2 4 6 8 10 12 Difference 1 1 1 1 1 1 So the formula is: 2n + 1 To check that the formula is correct we can try and put it into

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work