# Beyond Pythagoras

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Introduction

Beyond Pythagoras Pythagoras was a great mathematician who created theorems and one of his famous theorems was the "Pythagoras Theorem". You start with a right-angled triangle. The hypotenuse is labeled "c". The bottom of the triangle is "b" and the side of the triangle is labeled "a". Pythagoras Theorem says that in any right angled triangle, the lengths of the hypotenuse and the other two sides are related by a simple formula. So, if you know the lengths of any two sides of a right angled triangle, you can use Pythagoras Theorem to find the length of the third side: Algebraically: a2 + b2 = c2 The numbers 3, 4 and 5 satisfy the condition 9 + 16 = 25 Because 3x3=9 4x4=16 5x5=25 And so 9 + 16 = 25 I now have to find out if the following sets of numbers satisfy a similar condition of: (Shortest Side) 2 + (middle Side) 2 = (Longest side) 2 a) 5, 12, 13 a2 + b2 = c2 52 + 122 = 132 25 + 144 = 169 169 = 169 b) (7, 24, 25) a2 + b2 = c2 72 + 242 = 252 49 + 576 = 625 625 = 625 (3, 4, 5), (5, 12, 13) and (7, 24, 25) are called Pythagorean triples because they satisfy the condition, (Shortest side)2 + (Middle side)2 = (Longest Side)2 We know from the Pythagorean triples the shortest side is always an odd number. So far I have observed the following patterns: The shortest side length advances by two each time. Both the shortest and longest side lengths are always odd. The longest length is always one unit more than the middle length. I can immediately see the formula to get the shortest side length from the term number. The terms multiplied by two add one equals the shortest side length. Let Term number = Tn Let Shortest Side length = SL Tn ------ 2n + 1 = SL If term number = ...read more.

Middle

For instance if we choose (8,15,17), 6 is an even number and 10 is 2 more than the middle number, therefore we cannot say that (8,15,17) falls under this category. The formulas found out for this family will also not work for (8,15,17). We can use the formulas found to prove, that (8,15,17) does not work for. If n = 1, the values are (8,15,17); We will now substitute 1 for the formulas found: Shorter side = 2n + 1 = 2 (1) + 1 = 3 Middle Side = 2n2 + 2n = 2 (1)2 + 2 (1) = 2 + 2 = 4 Longest Side = (2n2 + 2n) + 1 = [ 2 (1)2 + 2 (1) ] + 1 = 4 + 1 = 5 We see that the values found are not (8,15,17) and we realize that the formulas are not meant for values where a is even and c is 2 more than b. If we choose another family where the n = 2, the values are (33,56,65), we see that a is an odd number and c is 9 more than b. We can use the formulas found to prove that (33,56,65) are not applicable to the first family. If n = 2, the values are (33,56,65); Shorter Side = 2n + 1 = 2 (2) + 1 = 4 + 1 = 5 Middle Side = 2n2 + 2n = 2 (2)2 + 2 (2) = 8 + 4 = 12 Longer Side = (2n2 + 2n) + 1 = [ 2(2)2 + 2 (2) ] + 1 = 12 + 1 = 13 Here again we see that the values found are not (33,56,65), and we realize that the formulas found are not meant for values where a is odd and c is 9 more than b. Now I would like to investigate other families of the Pythagorean triples, which I am able to do with the help of the site www.oswego.edu/multi-campus-nsf/ David-Dennis-mathed.htm My second ...read more.

Conclusion

I would like to establish a generalized formula for all the even families. Serial no. Family b c 1 b + 2 a2 - 22 2*2 a2 + 22 2*2 2 b + 4 a2 - 42 2*4 a2 + 42 2*4 3 b + 6 a2 - 62 2*6 a2 + 62 2*6 4 b + 8 a2 - 82 2*8 a2 + 82 2*8 5 b + 10 a2 - 102 2*10 a2 + 102 2*10 b + z a2 - z2 2*z a2 + z2 2*z Next, I would like to establish a generalized formula for all the odd families. Serial no. Family b c 1 b + 5 a2 - 52 2*5 a2 + 5 2*5 2 b + 9 a2 - 92 2*9 a2 + 92 2*9 3 b + 11 a2 - 112 2*11 a2 + 11 2*11 4 b + 15 a2 - 152 2*15 a2 + 152 2*15 5 b + 17 a2 - 172 2*17 a2 + 172 2*17 b + z a2 - z2 2*z a2 + z2 2*z Now, I would like to establish a generalized formula for all the square families. Serial no. Family b c 1 b + 4 a2 - 42 2*4 a2 + 42 2*4 2 b + 9 a2 - 92 2*9 a2 + 92 2*9 3 b + 25 a2 - 252 2*25 a2 + 252 2*25 4 b + 36 a2 - 36 2*36 a2 + 36 2*36 5 b + 49 a2 - 492 2*49 a2 + 49 2*49 b + z a2 - z2 2*z a2 + z2 2*z Through all these investigations done, it has helped me to know the Pythagoras theorem in much more detail. It has even helped me in investigating on various families, which are odd, even, etc. It has even helped me find formulas which I could apply to find out the various Pythagorean triples. 1 ...read more.

This student written piece of work is one of many that can be found in our GCSE Pythagorean Triples section.

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