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beyond pythagoras
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Maths GCSE Coursework Beyond Pythagoras Aim The aim of this coursework is to investigate the relationship between the lengths of the three sides of right angle triangles, the perimeters and the areas of the triangles. For any right-angled triangle the (shortest side) ²+ (middle side) ² will equal to the (longest side) ². I have represented this in the simple formula below. a ²+b ²=c ² The numbers 3, 4 and 5 satisfy the equation. This is because 3² + 4² = 5² = (3 x 3) + (4 x 4) = (5 x 5) 9 + 16 = 25 The overall formula for all Pythagorean triples is :- (shortest side) ²+(middle side) ²=(longest side) ² 1.a) I will use the set of numbers 5,12 and 13 to check this formula. (shortest side) ²+( middle side) ²=(longest side) ² 5² + 12² = 13² 25 + 144 = 169 This set of numbers is a Pythagorean triple as they follow the formula b) I am going to check if this set of numbers is also a triple 7,24 and 25. (shortest side) ²+(middle side) ²=(longest number) ² 7² + 24² = 25² 49 + 576 = 625 These numbers follow the formula also and so they are a Pythagorean triple. The perimeter for this right angle triangle is to add all of the three sides together:- 3+4+5= 12 units The formula for calculating the area of a right angle triangle is:- base x height 2 (3 x 4) =6 sq units 2 2a) I will now calculate the perimeter and area for the following triangles:- The perimeter for this triangle is :- The perimeter for this triangle is:- 5+12+13=30 units 7+24+25=56 units The area for this triangle is:- The area of this triangle is :- (5 x 12) =30 sq units (7 x 24) = 84 sq units 2 2 2b)These are the results of the perimeters and areas of the right angled triangles. Length of shortest side Length of middle side Length of longest side Perimeter (units) Area(square units) 3 4 5 12 6 5 12 13 30 30 7 24 25 56 84 I am going to investigate the relationshi betweent the perimeters, the areas, and the lengths of the sides. To start with I have placed the first seven pythagorean triples that I discovered with an odd number for the shortest side in a table with the value of n that will corresponds to the formulas worked out for each side length. The perimeters and area have also been calculated. I can use this table to check the answers for the formulas that I have worked out. n Shortest side Middle side Longest side Perimeter Area 1 3 4 5 12 6 2 5 12 13 30 30 3 7 24 25 56 84 4 9 40 41 90 180 5 11 60 61 132 336 6 13 84 85 182 546 7 15 112 113 240 840 Firstly I am going to work out the formulas for each of the columns in the table. Shortest Side I worked out that the formula for the shortest side is 2n+1 Method This is because there is a difference of two in each case. n 1 2 3 4 5 3 5 7 9 11 +2 +2 +2 +2 =2n + x when "n" is equal to three (2 x 3) +x =7 x =7 - (2 x 3) x =7 - 6 x =1 so the formula for the shortest side therefore is 2n +1.
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