Beyond Pythagoras Math Investigation
Pythagoras Theorem:
Pythagoras states that in any right angled triangle of sides 'a', 'b'
and 'c' (a being the shortest side, c the hypotenuse): a2 + b2 = c2
[IMAGE]
E.g. 1.
32 + 42= 52
9 + 16 = 25
52 = 25
2. 52+ 122= 132 3. 72 + 242 = 252
25 + 144 = 169 49 + 576 = 625
132 = 169 252 = 625
All the above examples are using an odd number for 'a'. It can
however, work with an even number.
E.g. 1. 102 + 242= 262
100 + 576 = 676
262 = 676
N.B. Neither 'a' nor 'b' can ever be 1. If either where then the
difference between the two totals would only be 1. There are no 2
square numbers with a difference of 1.
32 9
42 16
52 25
62 36
72 49
82 64
92 81
102 100
112 121
As shown in the above table, there are no square numbers with a
difference of anywhere near 1.
Part 1:
Aim: To investigate the family of Pythagorean Triplets where the
shortest side (a) is an odd number and all three sides are positive
integers.
By putting the triplets I am provided with in a table, along with the
next four sets, I can search for formulae or patterns connecting the
three numbers.
Pythagorean Triplet
(n)
1st Number
(a)
2nd Number
(b)
3rd Number
(c)
Area
(cm)
Perimeter
(cm)
1
2
3
4
5
6
7
3
5
7
9
11
13
15
4
12
24
40
[IMAGE] ½ (a2 + b2) times it by the ratio of its real area to a
Sum Law (the sum of the interior angles of a triangle must sum to 180
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This shows that there is a difference of 2cm between A and B, and B
will enable me to see a pattern in the shapes so I can make a table
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