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Beyond Pythagoras

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Introduction

Beyond Pythagoras

Introduction:

a² + b² = c²

The numbers 3, 4 and 5 can be the lengths of the sides of a right-angled triangle.

image00.pngimage01.png

image07.png

image03.png

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The perimeter = a + b + c           The area = a ×  b  ÷ 2

The numbers 5, 12 and 13 can also be the lengths of the sides of a right-angled triangle.

image16.png

image19.pngimage17.pngimage18.png

image20.pngimage02.png

                                        This is also true for 7, 24 and 25.image03.pngimage03.png

image04.pngimage05.png

These numbers are all called Pythagorean triples because they satisfy the condition.

Aim: I am going to investigate the different values of a, b and c for which the formula     a² + b² + c² works. I will also investigate the even and odd values for (a) for which the formula works.

...read more.

Middle

112

113

240

840

8

17

144

145

306

1224

Formulas:

The formula for a is:

N                            a

1                             3          = 1 ×2 + 1image08.png

2                       5          = 2× 2 + 1

3                       7          = 3 × 2 + 1

4                       9          = 4 ×2 + 1

5                      11               = 5 × 2 + 1image09.png

          n × 2 + 1

So from this I can work out that the general rule for A is 2n + 1.

Now I will try and work out a formula for B.

B

4

12

24

40

N

1

2

3

4image10.png

1

4

9

16

Difference between b and n

2

4

6

8

So the formula for B is n² + n × 2 + n²

C is one bigger than B so the formula for C is n²  + n × 2  + n²  + 1.

The formula for P = a + b + c so that is (2n + 1)+(n²  + n × 2 + n² )+(n²  + n × 2 +n² + 1) = P

For example 7 + 24 + 25  = 56

The formula for area is A × B  ÷  2 so that means (2n + 1)(n²   +  n ×

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Conclusion

B

8

15

24

35image13.png

N

1

2

3

4

1

4

9

16

Difference between b and n

3

8

15

19

So the formula for B is n²  + 4n + 3

C is two bigger than B so the formula for C is n² + 4n + 3  + 2.

The formula for P = a + b  + c so that is (2n + 4) + (n² + 4n + 3 ) + (n² + 4n + 3  + 2) = P .

For example 6 + 8 + 10 =  24

The formula for area is A ×  B ÷ 2 so that means (2n + 4)(n²  +  4n + 3) ÷ 2.

For example 6 × 8 ÷ 2 = 24

Part 3:

I will now change the rules again:

  1. (a) Must be a multiple of 3
  2. (b) Must be a multiple of 4
  3. (c) Must be a multiple of 5

I will also try out my formulas from my previous Pythagorean triples and see if they work.

N

Aimage06.png

Bimage06.png

Cimage06.png

Pimage06.png

Areaimage06.png

1

3    

4

5

12

6

2

6

8

10

24

24

3

9

12

15

36

54

4

12

16

20

48

96

5

15

20

25

60

150

image14.png

Hi h

...read more.

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