Beyond Pythagoras.

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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

22 = 12 x 12 = 144

32 = 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 = 1/2 x 3 x 4 = 6

5, 12, 13

Perimeter = 5 + 12 + 13 = 30

Area = 1/2 x 5 x 12 = 30

7, 24, 25

Perimeter = 7 + 24 + 25 = 56

Area = 1/2 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'
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• 'b' increases by +4 each term

• ('a' x 'n') + n = 'b'

From these observations I have worked out the next two terms.

I will now put the first five terms in a table format.

Term Number 'n' Shortest Side 'a' Middle Side 'b' Longest Side 'c' Perimeter

Area

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

I have worked out formulas for
...

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