• 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

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 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

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

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

/

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

Related GCSE Pythagorean Triples essays

1. Beyond Pythagoras

This means the equation is something to do with 2n. So, I will write down all the answers for 2n. 2, 4, 6, 8, 10, 12 +1 +1 +1 +1 +1 There is only a difference of +1 between 2n and the shortest side, so this means the formula should be 2n+1.

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 am now going to see if there is a formula to work out the perimeter just using N. The easiest thing to say would just be to work out the perimeter using the previous three equations. You could just say a + b + c.

2. 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

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

I shall now try adding n to the formula to make the equation 2n� + n. 2 x 1� + 1 = 3 2 x 2� + 2 =10 2 x 3� + 3 = 21 I have noticed that each of these results is n less than the number I was aiming to get.

2. Beyond Pythagoras

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.

1. Maths GCSE coursework: Beyond Pythagoras

the sequence: n = 3 the sequence number for this is 7 . 2n + 1 2 x 3 + 1 6 + 1 = 7 It is correct! To see if there are any more formulae in the table we will first look at the 'middle number' sequence: 4

2. Beyond Pythagoras .

that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules.

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