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 Pythagoras Theorem is a² + b² = c². 'a' being the shortest side, 'b' being the middle side and 'c' being the longest side (which is always the 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 We also checked to see if similar sets of numbers also satisfy this condition: (smallest number)² + (middle number)² = (largest number)² The numbers 5, 12 and 13 also satisfy this condition: 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 The numbers 7, 24 and 25 also satisfy this condition: 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 For the set of numbers 3, 4 and 5: Perimeter = 3 + 4 + 5 = 12 Area = 1/2 x 3 x 4 = 6 For the set of numbers 5, 12 and 13: Perimeter = 5 + 12 + 13 = 30 Area = 1/2 x 5 x 12 = 30 For the set of numbers 7, 24 and 25: Perimeter = 7 + 24 + 25 = 56 Area = 1/2 x 7 x 24 = 84 From these sets of numbers I have noticed the following: - * 'a' increases by +2 for each set of numbers * 'b' increases by +4 for each set of numbers * 'c' is
3 Digit Number - Maths Investigations
CWK1 3 Digit Number Take any 3 digit number, write down all possible numbers that can be made with the three digits, add them up, divide the total by the sum of the 3 digits. Investigate. If All Digits Are Different : - 123 456 789 132 465 798 213 546 879 231 564 897 321 654 987 +312 +645 +978 332 ? 24=222 3330 ? 15=222 5328 ? 24=222 147 258 369 174 285 396 471 528 639 417 582 693 741 825 936 +714 +852 +963 2664 ? 12=222 3330 ? 15=222 3996 ? 18=222 It seems that when all 3 digits are different the answer to the problem is 222. Can I use Algebra to explain this? abc=100a+10b+c acb=100a+10c+b bac=100b+10a+c bca=100b+10c+a cba=100c+10b+a +cab=100c+10a+b 222a+222b+222c = 222(a+b+c) a+b+c =222 What if 2 of the 3 digits are the same? If 2 digits are the same : - 223 334 566 322 343 656 +232 +433 +665 777 ? 7=111 1110 ? 10=111 1887 ? 17=111 224 559 772 242 595 727 +422 +955 +277 888 ? 8=111 2109 ?19=111 1776 ? 16=111 It seems that when 2 of the 3 digits are the same the answer to the problem is 111. Can I use Algebra to explain this? aab=100a+10a+b aba=100a+10b+a +baa=100b+10a+a 222a+111b = 111(2a+b) 2a+b =111 What if all 3 digits are the same number? If All Digits Are The Same : - 333
Math's Coursework: Pythagoras triples.
Math's Coursework: Pythagoras triples. I am investigating the relationship of the sides in a Pythagoras triple. I will hopefully be able to find formulas for each side, and the perimeter and Area. To do this I must first discover the associations each side of a Pythagorean triples have (the shortest, middle and longest side.) A Pythagorean triples are basically the sides of a right-angle triangle according to Pythagoras. The formula is: a + b= c Length of Shortest side ( a ) Length of middle side ( b ) Length of longest side ( c ) Perimeter Area 3 4 5 2 6 5 2 3 30 30 7 24 25 56 84 This is a table of the sides, perimeter and area of the first three Pythagorean triples. Judging by the table I can recognize a few of relationships between the numbers: * The shortest and longest sides are odd numbers. * The value of the middle side is the longest side's value -1. * The lengths of the shortest sides are 2 more then the previous shortest side length. I'm going to extend the table with the wish that I find more patterns and figure out a formula. 9 40 41 90 80 1 60 61 32 330 Patterns of the Middle Value side. 4 8 12 4 12 24 4 16 40 4 20 60 I have found that the middle value has a difference of four. All of the middle values are devisable by 4; I'm going to divide all of middle values by four, then see what further patterns I
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 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