• 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
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17
18. 18
18
19. 19
19
20. 20
20
21. 21
21
22. 22
22
23. 23
23
24. 24
24
25. 25
25
26. 26
26
27. 27
27
28. 28
28
29. 29
29
30. 30
30
31. 31
31
32. 32
32
• Level: GCSE
• Subject: Maths
• Word count: 1600

# Beyond Pythagoras - I am investigating the relationships between the lengths of the three sides of right angled triangles, the perimeters and areas of these triangles.

Extracts from this document...

Introduction

CONTENTS

Introduction....................................................2

Proof of Pythagoras’ Thereom........................4

Prediction........................................................6

Workings.........................................................7

The Table of Results......................................12

Workings.......................................................13

The Table of Results......................................18

nth term for ‘length of shortest side’.............19

nth term for ‘length of middle side’...............21

nth term for ‘length of longest side’..............23

nth term for ‘perimeter’................................25

nth term for ‘area’.........................................27

Pattern.........................................................28

The End........................................................29

Investigation: Beyond Pythagors

Introduction

I am investigating the relationships between the lengths of the three sides of right angled triangles, the perimeters and areas of these triangles.

I was set to predict about Pythagorean triples, make generalisations about the lengths of sides and make generalisations about the perimeter and area of corresponding triangles.

Pythagorean triples or triad are a set of three positive integers (a, b and c) they are representing the sides of a triangle and satisfying Pythagoras’ thereom (a + b = c )

Let me tell you more about Pythagoras’ thereom.                                                                             Pythagoras was a greek philosopher and mathematician who lived in the sixth century BC.

Middle

Perimeter= 5+ 12+13= 30 units

Area= ½ x 5 x 12= 30 square units

13

5

12

3

Perimeter= 7+ 24+ 25= 56 units

25                   Area= ½ x 7 x 24= 84 square units

7

24

4

Perimeter= 9+ 40+ 41= 90 units

Area= ½ x 9 x 40= 180 square units

41                                      9

40

5

Perimeter= 11+ 60+ 61= 132 units

61       Area= ½ x 11 x 60= 330 square units

11

60

6

Perimeter= 13+ 84+ 85= 182 units

Area= ½ x 13 x 84= 546 square units         85

13

84

7

Perimeter=15+ 112+ 113= 240 units

113       Area= ½ x 15 x 112= 840 square units

15

112

8

Perimeter= 17+ 144+ 145= 306 units

Area= ½ x 17 x 144= 1224 square units

145

17

144

9

Perimeter= 19+ 180+ 181= 380 units

Area= ½ x 19 x 180= 1710 square units

19                     181

180

10

Perimeter= 21+ 220+ 221= 462 units

Area= ½ x 21 x 220= 2310 square units

221

21

21

220

11

Perimeter= 23+ 260+ 261= 544 units

Area= ½ x 23 x 260= 2990 square units

65                             33

56

12

Perimeter= 25+ 304+ 305= 634 units

Conclusion

2 x 3(3+1) =24

2 x 4(4+1) =40

2 x 5(5+1) =60

2 x 6(6+1) =84

2 x 7(7+1) =112

2 x 8(8+1) =144

2 x 9(9+1) =180

2 x 10(10+1) =220

etc.

nth term for ‘Length of longest side’

5, 13, 25, 41, 61, 85, 113, 145, 181, 221...

8   12   16  20  24  28    32    36    40

4     4     4     4    4      4      4      4

2n  2  8  18  32  50  72  98  128  162  200...

1    3  5  7  9  11  13  15  17  19  21

2   2  2  2   2    2    2   2     2     2

nth term= 2n(n+1)+1

Check

2 x 1(1+1) +1 =5

2 x 2(2+1) +1 =13

2 x 3(3+1) +1 =25

2 x 4(4+1) +1 =41

2 x 5(5+1) +1 =61

2 x 6(6+1) +1 =85

2 x 7(7+1) +1 =113

2 x 8(8+1) +1 =145

2 x 9(9+1) +1=181

2 x 10(10+1) +1 =221

etc.

nth term of ‘Perimeter’

12, 30, 56, 90, 132, 182 240, 306...

18  26  34  42     50    58    66

8     8     8     8        8      8

4n    4  16  36  64  100  144  196  256...

28  14  20  26  32  38  44  50

6    6    6    6    6     6     6   6

nth term= 4n + 6n+ 2

Check

(4 x 1)+(6 x 1)+2 =12

(4 x 2)+(6 x 2)+2 =30

(4 x 3)+(6 x 3)+2 =56

(4 x 4)+(6 x 4)+2 =90

(4 x 5)+(6 x 5)+2 =132

(4 x 6)+(6 x 6)+2 =182

(4 x 7)+(6 x 7)+2 =240

(4 x 8)+(6 x 8)+2 =306

etc.

nth term of ‘Area’

6, 30, 84, 180, 330, 546, 840...

24  54  96   150   216  294

30   42  54    66     78

12    12   12   12

6n     6  24  54  96  150  216  294...

00    6    30    84    180    330    546

0    6     24    54    96    150     216

6    18    30    42    54       66

12   12    12    12      12

nth term= 6n + 12n

Check

(6 x 1) + (12 x 1) =6

(6 x 2) + (12 x 2) =30

(6 x 3) + (12 x 3) =84

(6 x 4) + (12 x 4) =180

(6 x 5) + (12 x 5) =330

(6 x 6) + (12 x 6) =546

(6 x 7) + (12 x 7) =840

etc.

Pattern

The pattern for the length of the shortest side, a, is that it is and will always be odd and prime numbers.

The pattern for the length of the middle side, b, is always odd numbers. The difference between the numbers are not equal but the difference of that is always +4. That is why the nth term is 2n(n+1).

The pattern for the length of the longest side, c, is one plus the middle side, b, the nth term is 2n(n+1)+1. The numbers could be odd or even.

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

1. ## Beyond Pythagoras

I'm going to do it with the third term: (2x3 +1) �+ (2x3 (3+1) ) � = (2x3 (3+1) +1)� 7�+24�=25� 49+576=625 ? 625=625, which is correct. Perimeter: P=a+b+c Which, once my formulas are substituted in, becomes: 2+6n+4n2 2+ (6x1)

2. ## Beyond Pythagoras

Therefore it is defiantly a Pythagoras triangle. Triangle 5 A2 + B2 = C2 112 + 602 = C2 121 + 3600 = C2 3721 = C2 V3721 = C2 61 = Side C Again this is a Pythagoras triangle. Triangle 6 A2 + B2 = C2 132 + 842 = C2 169 + 7056 = C2

1. ## Beyond Pythagoras

you get 2, now I just need to put this behind an n (2n). So if I add this onto the end of the 2n2 I got earlier I get 2n + 2n2 which I believe is the formula for this side of family 1!!!!!!!

2. ## Beyond Pythagoras

b� = (n�+4n+3)� ) c� = (n�+4n+5)� = a� = (2n+4) (2n+4) 4n�+8n+8n+16 4n�+16n+16 b� = (n�+4n+3) (n�+4n+3) n +4n�+3n�+4n�+16n�+4n+12n+3n�+12n+9 n +8n�+22n�+24n+9 a�+b� = n + 8n�+26n�+40n+25 c� = (n�+4n+5) (n�+4n+5) n + 4n�+5n�+ 4n�+16n�+20n+5n�+20n+25 n + 8n�+26n�+40n+25 I will give an example of how to use the formulae for

1. ## Beyond Pythagoras

and the first term is 10 so a difference is +2 If n=3 12 and the first term is 14 so a difference is +2 Therefore the rule for finding the nth term of the shortest side is: 4n+2 Mahmoud Elsherif Beyond Pythagoras P.9 Next I shall find the nth term of the middle side.

2. ## Beyond Pythagoras ...

2 - (2n+2n2)2 Shortest 2n+1+ 2n+1 4n2+ 2n 2n + 1 4n+ 4n2+1 Middle 2n2+2n + 2n2+2n 4n4+4n3 4n3+ 4n2 4n2 +8n3+4n4 Longest 2n2+2n+1 + 2n2+2n+1 4n4+4n3+2n2 4n3+ 4n2+2n 2n2+2n+1 4n4+8n3+8n2+4n+1 (4n+ 4n2+1)= (4n4+8n3+8n2+4n+1)-(4n2 +8n3+4n4) Now I have finished that I will start having even numbers, to see if Pythagoras's Theorem works.

1. ## Beyond Pythagoras

one and can be expressed as the formulae: an + a= b Relations between 'b' and 'c': The obvious relation here is that 'c' is always one more than 'b'. E.g. 5 is one more than 4 13 is one more than 12 and so on...

2. ## Pythagoras [Samos, 582 - 500 BC].

Men and women in the society were treated equally -an unusual thing at the time- and all property was held in common. Members of the society practiced the master's teachings, a religion the tenets of which included the transmigration of souls and the sinfulness of eating beans.

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