# Beyond Pythagoras

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Introduction

Beyond Pythagoras The aim of this investigation is to investigate Pythagoras theorem and to find a formula for the shortest side, middle length, hypotenuse, area and perimeter. Because I have typed this up on a computer... ^ is squared * is times / is divided Pythagoras is A2 + B2 = C2 I am going to prove this theory be finding out if the following numbers adhere to the rule. Triangle 1 5 12 13 5^ + 12^ = 13^ 5^ = 5*5 = 25 12^ = 12*12 = 144 13^ = 13*13 = 169 So 5^ + 12^ = 25 + 144 = 169 = 25^ The perimeter of the triangle is All the lengths of the side added up 5 + 24 + 25 = 30 The area of the triangle is 1/2 base * height 1/2 * 12 * 5 = 30 Triangle 2 7 24 25 7^ + 24^ =25^ 7^ = 7*7 = 49 24^ = 24*24 = 576 25^ = 25*25 = 625 So 7^ + 24^ = 49 + 576 = 625 = 25^ The perimeter of the triangle is All the lengths of the side added up 7 + 24 + 25 = 56 The area of the triangle is 1/2 base * height 1/2 * 24 * 7 = 84 Length of shortest side Length of middle side Length of longest side Perimeter Area 3 4 5 ...read more.

Middle

and the length of the hypotenuse. I will also work out a formula for the area and the perimeter of the triangle. By taking the date from the "beyond Pythagoras "sheet and the data on the excel sheet I can work out the formula by using a a sequence diagram to find a constant and then using a linear equation if it there is a constant on the second step or a quadratic if there is a constant on the third step. I this is how I will work out the formula for the shortest side of family 1 X Y 1 3 > 2 2 5 > 2 3 7 There is a constant on the second step so I can solve it with a linear equation. I will solve it using the standard linear equation Y = ax + c I will put the numbers into this equation so I can work out the formula. A * 1 + c = 3 (i) A * 2 + c = 5 (ii) I will then take "ii" from "i" to find out what a is = 2 I will then put this into "i" to find out what "c" is 2 + c = 3 (i) I have to take 2 from three C = 1 Y = 2 * x + 1 X is the sequence number Checks 2 * 1 + 1 = 3 ...read more.

Conclusion

2 * x + 1 + Y = 2x2 + 2x + Y = 2x2 + 2x + 1 = (c)nick Peplow Y = 4x2 + 6x + 2 It will give the perimeter. Checks Y = 4 * 12 + 6 * 1 + 2 = 12 Y = 4 * 22 + 6 * 2 + 2 = 30 Y = 4 * 32 + 6 * 3 + 2 = 56 By checking I have found that my formula is correct. Now I need to find the area The formula for finding the area of a triangle is 1/2 base * height So in need to divide the formula for "b" by two and then times it by the formula for "a" This is the formula for "b" Y = 2x2 + 2x (i) And this is the formula for "a" Y = 2 * x + 1 (ii) So basically all you have to do is divide the formula that you get from "i" And then times it by "ii" Y = (2x2 + 2x) /(divided) 2 * (2x + 1) Check Y = (2*12 + 2*1) / 2 * (2*1 + 1) =6 Y = (2*22 + 2*2) / 2 * (2*2 + 1) =30 Y = (2*32 + 2*3) / 2 * (2*3 + 1) =84 By checking my formula I have proved that it is correct. (c)nick Peplow (c)Nick Peplow ...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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