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GCSE: Pythagorean Triples
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- Level: GCSE
- Questions: 75
1; 2(1)+1=3 If term number = 2; 2(2)+1=5 If term number = 3; 2(3)+1=7 This works for all Pythagorean triples that have an odd numbered SL. I have investigated another three Pythagorean triples and they are stated below. Serial No. Shortest side a Middle Side b Longest side c 1 3 4 5 2 5 12 13 3 7 24 25 4 9 40 41 5 11 60 61 6 13 84 85 I will now test the Pythagorean triples I have just found with the aid of Pythagoras theorem and a diagram.
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Investigate the area of triangle studies including the Pythagorean Theorem and in particular Pythagorean Triples, sets of numbers where the shortest side is an odd value and all three are positive whole integers.
This means that the equation I want has nothing to do with 3 sides squared, and I can eliminate this from my list of sides to investigate. I will now try 2 sides squared. (Middle)� + Largest number = (smallest number)� = 122 + 13 = 52 = 144 + 13 = 25 = 157 = 25 This does not work and I know that neither will 132, because it is larger than 122. I have decided that there is also no point in squaring the largest and the smallest or the middle number and the largest number.
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They are always consecutive numbers. I shall now investigate this. I will assume that the hypotenuse has length b + 1 where b is the length of the middle side. (c=b+1) b+1 a b Using Pythagoras: a? + b? = (b+1) ? Expanding this: a? + b? = b? + 2b+ 1 therefore a? = 2b +1 This means that a? must be odd because it equals 2b + 1. Since a? is odd this means that a must be odd also. (even x even = even)
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They are always consecutive numbers. I shall now investigate this. I will assume that the hypotenuse has length b + 1 where b is the length of the middle side. (c=b+1) b+1 a b Using Pythagoras: a� + b� = (b+1) � Expanding this: a� + b� = b� + 2b+ 1 therefore a� = 2b +1 This means that a� must be odd because it equals 2b + 1. Since a� is odd this means that a must be odd also. (even x even = even)
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but this is a part of another pattern which is that the square of the shortest side is the same as the middle and longest sides added together, So... a� = b + c The theory works because: 12 +13 = 25 = 5� 5� = 12 + 13 This now allows to predict, odd starting, Pythagorean triples. It is correct! We have to, now, look at the smallest number. 3 5 7 9 11 13 2 2 2 2 2 1st difference The 1st difference is 2 so we must find the formula for the 'smallest number'.
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I also knew that c was probably one larger than b. I tried multiple triples until I found one that seemed to agree the theorem. I came up with 9, 40, 41. I proved that this was right: 92+402= 412 Because 92= 9x9= 81 402= 40x40= 1600 412= 41x41= 1681 81+1600= 1681 Next I needed to find the perimeter and area for this triple. I used the formula: Perimeter= a+b+c P= 9+40+41 P= 90 I found out the area using the formula: Area= (axb)?2 A= (9x40)?2 A=360?2 A=180 Next I updated my table: Length of Shortest Side (a)
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4 +5 = 32 9 = 9 And... 24 + 25 = 72 49 = 49 It works with both of my other triangles. So... Middle number + Largest number = Smallest number2 If I now work backwards, I should be able to work out some other odd numbers. E.g. 92 = Middle number + Largest number 81 = Middle number + Largest number I know that there will be only a difference of one between the middle number and the largest number. So, the easiest way to get 2 numbers with only 1 between them is to divide 81 by 2 and then using the upper and lower bound of this number.
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Therefore, the first odd Pythagorean triple to satisfy these criteria is the 3,4,5 triple, and it is subsequently the first term of the odd triples sequence. The first sequence of triples that will be investigated are those with the first number (i.e. 3 of 3,4,5) as an odd number. I will refer to these triples as "odd triples" throughout the investigation for convenience. The second sequence of triples that will be investigated are those with the first number (i.e. 6 of 6,8,10, although this triple should not be counted as a "true" triple, as we shall see later)
- Word count: 4016