Beyond Pythagoras
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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 = x 3 x 4 = 6
5, 12, 13
Perimeter = 5 + 12 + 13 = 30
Area = x 5 x 12 = 30
7, 24, 25
Perimeter = 7 + 24 + 25 = 56
Area = 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'
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
1
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
How to get 'a' from 'n'
How to get 'b' from 'n'
How to get 'c' from 'n'
How to get the perimeter from 'n'
How to get the area from 'n'
My formulas are
2n + 1
2n2 + 2n
2n2 + 2n + 1
4n2 + 6n + 2
2n3 + 3n2 + n
To get these formulas I did the following
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
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
Side 'c' is just the formula for side 'b' +1
The perimeter = a + b + c. Therefore I took my formula for 'a' (2n + 1), my formula for 'b' (2n2 + 2n) and my formula for 'c' (2n2 + 2n + 1)...