#### BEYOND PYTHAGORAS

BEYOND PYTHAGORAS In this report, I am going to investigate the Pythagorean Triple. The Pythagoras Theorem was invented by Pythagoras, a Greek mathematician and philosopher who lived in the 6th centaury BC. The Pythagoras Theorem only works in right-angled triangles, where there are three different lengthed sides, one short, one medium, and the other long. A Pythagorean Triple is when a set of numbers satisfy the condition: Shortest side2 + Medium side2 =Longest side2. Also all the sides have to have positive integers. Here is an example of a Pythagorean Triple: ) The above triangle is a Pythagorean Triple because it satisfies the condition with all its sides being a positive integer. I will now work out the perimeter and area of the above Pythagorean Triple. There are also other Pythagorean Triples. Here they are: 2) 3) Both the triangles 2) and 3) are Pythagorean Triples because they satisfy the condition and all their sides have a positive integer. Here is a table showing the results of the 3 Pythagorean Triples: Triangle No. Shortest side Medium side Longest side Perimeter Area ) 3 4 5 2 62 2) 5 2 3 30 302 3) 7 24 25 56 842 From the above table, I can see a few patterns emerging. Here they are: i. The shortest side is always an odd number. ii. The medium side is always an even number. iii. The medium side plus one equals to the

#### Beyond Pythagoras

Beyond Pythagoras Introduction: During this investigation I will be trying to find out patterns and formulas relating to pythagorus' theorem. One pattern he found with triangles was that the smallest and middle length sides squared added together to make the largest side squared. For example: 3 + 4 = 5 because 3 = 3 x 3 = 9 4 = 4 x 4 = 16 5 = 5 x 5 = 25 so 3 + 4 = 9 + 16 = 25 = 5 (smallest number) + (middle number) = (largest number) Another name for this is Pythagorean triples: a + b = c I will continue with this investigation to find as many rules and formulas as possible, to see if this is a one off, or if it only occurs in certain triangles. Further into my investigation I will also look at triangles that don't fit into the rules I've found, but whose smallest and middle length side when squared do add up to the longest length side squared. I began by checking to see if these triangles fit into pythagoras' triangle theorem: a) 5, 12, 13 b) 7, 24, 25 Both fit into the pattern. The numbers, 3, 4, 5 could be used to make a right angled triangle as shown below: The perimeter and area of this triangles can be worked out as follows: * Perimeter = 3 + 4 + 5 =12 units or smallest length + middle length + largest length = perimeter in appropriate unit. e.g. 3 cm + 4 cm + 5 cm = 12 cm perimeter * Area = 1/2 x 3 x 4 = 6 square

#### Beyond Pythagoras

Beyond Pythagoras Introduction: a² + b² = c² The numbers 3, 4 and 5 can be the lengths of the sides of a right-angled triangle. The perimeter = a + b + c The area = a × b ÷ 2 The numbers 5, 12 and 13 can also be the lengths of the sides of a right-angled triangle. This is also true for 7, 24 and 25. These numbers are all called Pythagorean triples because they satisfy the condition. Aim: I am going to investigate the different values of a, b and c for which the formula a² + b² + c² works. I will also investigate the even and odd values for (a) for which the formula works. I also intend to find out a relationship between a, b and c and their perimeters and areas. The rules for my first Pythagorean triples are: . (a) Must be a odd consecutive number 2. (b) Must be one smaller than c 3. (c) Must be one bigger than b I am now going to construct a table for these Pythagorean triples and I will try and work out patterns and formulas to help me work out the value of a, b and c and the perimeter and area of the triangles. (n) (a) (b) (c) (p) (Area) 3 4 5 2 6 2 5 2 3 30 30 3 7 24 25 56 84 4 9 40 41 90 80 5 1 60 61 32 330 6 3 84 85 82 546 7 5 12 13 240 840 8 7 44 45 306 224 Formulas: The formula for a is: N a 3 = 1 ×2 + 1 2 5 = 2× 2 +

#### Beyond Pythagoras

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 69 = 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

#### Beyond Pythagoras

Beyond Pythagoras Introduction Believed to have been born in 582BC and died in 500BC approximately, Pythagoras was a Greek philosopher and mathematician. He discovered some of the most influential theories of number, geometry and proportion which are still frequently used in modern mathematics. Pythagorean triples, on which this investigation is based, are sets of three numbers as seen in the above diagram. Here it is the simplest triple; 3,4,5. They will be written in the following form throughout this paper; a,b,c. There are other Pythagorean triples such as 5,12,13; which is the 2nd odd triple and 361,65160,65161; which is the 180th odd triple. His theorem is this: the square of the hypotenuse (longest side (c (5 (25)))) of a right-angled triangle is equal to the sum of the squares of the other two sides (a (3 (9)) and b (4 (16))). This can be expressed using the following equation: a2+b2=c2 Aim The aim of the investigation is to explore and mathematically express the relationships between the number of the triple (n), the length of each side (a; the shortest), (b; the intermediate), (c; the longest), the perimeter (P) and area (A) of a right-angled triangle. Please note: Throughout this investigation, the sum on the bottom line is the final answer to each set of sums. For example: y=2+2 y=4 In this example, the bottom line (y=4) should be interpreted as

#### Pythagorean Triples

Pythagorean