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  • 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

image11.jpgimage00.pngimage01.png

image12.png

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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.image14.png

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.

...read more.

Middle

Perimeter= 5+ 12+13= 30 units

Area= ½ x 5 x 12= 30 square units                

                                                                      13

                                                                                                           5

image09.png

                                                                                12

3  

image10.png

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

...read more.

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.

...read more.

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