• 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
• Level: GCSE
• Subject: Maths
• Word count: 1235

# Beyond Pythagoras.

Extracts from this document...

Introduction

## Beyond Pythagoras

Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (hypotenuse) of a right angled triangle.

The numbers 3, 4 and 5 satisfy this condition

32 + 42 = 52

because 32 = 3 x 3 = 9

42 = 4 x 4 = 16

52 = 5 x 5 = 25

and so 32 + 42 = 9 + 16 = 25 = 52

The numbers 5, 12, 13 and 7, 24, 25 also work for this theorem

52 + 122 = 132

because 52 = 5 x 5 = 25

122 = 12 x 12 = 144

132 = 13 x 13 = 169

and so 52 + 122 = 25 + 144 = 169 = 132

72 + 242 = 252

because 72 = 7 x 7 = 49

242 = 24 x 24 = 576

252 = 25 x 25 = 625

and so 72 + 242 = 49 + 576 = 625 = 252

3 , 4, 5

Perimeter = 3 + 4 + 5 = 12

Area = ½ x 3 x 4 = 6

5, 12, 13

Perimeter = 5 + 12 + 13 = 30

Area = ½ x 5 x 12 = 30

7, 24, 25

Perimeter = 7 + 24 + 25 = 56

Area = ½ x 7 x 24 = 84

From the first three terms I have noticed the following: -

• 'a' increases by +2 each term
• 'a' is equal to the term number times 2 then add 1
• the last digit of 'b' is in a pattern 4, 2, 4
• the last digit of 'c' is in a pattern 5, 3, 5
• the square root of ('b' + 'c') = 'a'
• 'c' is always +1 to 'b'
• 'b' increases by +4 each term
• ('a' x 'n') + n = 'b'

Middle

60

61

132

330

I have worked out formulas for

1. How to get 'a' from 'n'
2. How to get 'b' from 'n'
3. How to get 'c' from 'n'
4. How to get the perimeter from 'n'
5. How to get the area from 'n'

My formulas are

1. 2n + 1
2. 2n2 + 2n
3. 2n2 + 2n + 1
4. 4n2 + 6n + 2
5. 2n3 + 3n2 + n

To get these formulas I did the following

1. Take side 'a' for the first five terms 3, 5, 7, 9, 11. From these numbers you can see that the formula is 2n + 1 because these are consecutive odd numbers (2n + 1 is the general formula for consecutive odd numbers) You may be able to see the formula if you draw a graph
1. From looking at my table of results, 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
1. Side 'c' is just the formula for side 'b' +1
2. The perimeter = a + b + c. Therefore I took my formula for 'a' (2n + 1)

Conclusion

4n2 + 6n + 2

4(n +3)2 - 9 + 2

4(n +3)2 - 7

4(n +3)2 = 7

(n +3)2 = 1.75

n + 3 = 1.322875656

n + 3 = -1.322875656

n = -1.677124344

n = -4.322875656

## Arithmatic Progression

I would like to know whether or not the Pythagorean triple 3,4,5 is the basis of all triples just some of them.

To find this out I have been to the library and looked at some A-level textbooks and learnt 'Arithmatic Progression'

3, 4, 5 is a Pythagorean triple

The pattern is plus one

If a = 3 and d = difference (which is +1) then

3 = a

4 = a + d

5 = a +2d

a, a +d, a + 2d

Therefore if you incorporate this into Pythagoras theorem

a2 + (a + d)2 = (a + 2d)2

a2 + (a + d)(a + d) = (a + 2d)2

a2 + a2 + ad + ad + d2 = (a + 2d)2

2a2 + 2ad + d2 = (a + 2d)2

2a2 + 2ad + d2 = (a + 2d)(a + 2d)

If you equate these equations to 0 you get

a2 - 3d2 - 2ad = 0

Change a to x

x2 - 3d2 - 2dx = 0

Factorise this equation to get

(x + d)(x - 3d)

Therefore

x = -d

x = 3d

x = -d is impossible as you cannot have a negative dimension

a, a+d, a + 2d

Is the same as

3d, 4d, 5d

This tells us that the only Pythagorean triples are 3, 4, 5 or multiples of 3, 4, 5 e.g. 6, 8, 10 or 12, 16, 20 etc.

This student written piece of work is one of many that can be found in our GCSE Fencing Problem 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 Fencing Problem essays

1. ## Beyond Pythagoras

529 C=265 264� = 69696 A=23 265� =70225 529 + 69696 = 70225 B=264 Area = 23 x 264 = 6072 2 = 3036 cm� Perimeter = 23 + 264 + 265 = 552cm 25� = 625 C=313 312� = 97344 A=25 313� = 97969 625 + 97344 = 97969

2. ## Beyond pythagoras - First Number is odd.

This simplifies down to 4n2+6n+2. Perimeter=4n2+6n+2 Area: The area = (a x b) divided by 2. Therefore I took my formula for 'a' (2n + 1) and my formula for 'b' (2n2 + 2n). I then did the following: - (2n + 1)(2n2 + 2n)

1. ## Beyond Pythagoras

explains that for the terms 1,2,3, the difference between the lengths of the shortest sides 3,5,7 is 2. This means that to get the new shortest sides for the new triples, I will add 2 each time. By doing so I have got: 7 + 2 = 9; 9 + 2 = 11 and 11 + 2 = 13.

2. ## Beyond Pythagoras

Middle side (b) Longest side (c) Area (A) Perimeter (P) 1 2cm 0cm 2cm 0cm2 4cm 2 4cm 3cm 5cm 6cm2 12cm 3 6cm 8cm 10cm 24cm2 24cm 4 8cm 15cm 17cm 60cm2 40cm 5 10cm 24cm 26cm 120cm2 60cm 6 12cm 35cm 37cm 210cm2 84cm 7 14cm 48cm 50cm 336cm2 112cm 8 16cm 63cm 65cm 504cm2 144cm

1. ## Beyond Pythagoras.

I know this works because when n = 0, an� + bn also is 0, this then means that "c" is the only number contributing to the answer. e.g. 8 - 4 = 4 4 - 4 = 0 c = 0 Now I can calculate "b" by substituting "a" and "c" for the values which have now been found.

2. ## Beyond Pythagoras.

4 + 1 = 5 Term 3 'a' = 2n + 1 = 2 � 3 + 1 = 6 + 1 = 7 Side 'b' I used quadratic rule to find the formula for this side: 1 diff = 8 The first difference would be: difference of 1st sequence

1. ## Pythagoras Theorem is a2 + b2 = c2. 'a' being the shortest side, 'b' ...

odd numbers (2n + 1 is the general formula for consecutive odd numbers) You may be able to see the formula if you draw a graph 2. From looking at my table of results, I noticed that 'an + n = b'.

2. ## Beyond Pythagoras.

But I don't know the formula so I will have to work that out. Firstly I will be finding out the formula for the shortest side. Length - 3 5 7 9 11 1st difference - 2 2 2 2 The differences between the lengths of the shortest side are 2.

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