# Beyond Pythagoras - Sample Essay

Introduction I am investigating the relationships between the lengths of the three sides of right angled triangles, the perimeters and areas of these triangles. I was set to predict about Pythagorean triples, make generalisations about the lengths of sides and make generalisations about the perimeter and area of corresponding triangles. Pythagorean triples or triad are a set of three positive integers (a, b and c) they are representing the sides of a triangle and satisfying Pythagoras’ thereom (a + b = c ) Let me tell you more about Pythagoras’ thereom.

Pythagoras was a greek philosopher and mathematician who lived in the sixth century BC. He stated a thereom that states that in a right-angled triangle, the square of the hypotenuse (the longest side of a right-angled triangle, which is always the opposite side of the right angle ) is equal to the sum of the squares of the two sides, opposite and adjacent. A a + b = c AC = AB + BC B C Pythagoras’ thereom helps us to solve the lights on a right-angled triangle. Method Step 1: Square them Square the two numbers, the sides of the triangle, you are given.

Step 2: Add or Subtract To find the longest side, you add the two squared number. To find the shorter side, you subtract the smaller squared number from the larger one. Step 3: Square root After adding or subtracting, take the square root. Then check if the answer is correct. Example: t t= 5 + 7 = 74 5 = 25+ 49 = 8. 6 cm (1 d. p) = 25+ 49 (add to find the 7 = 74 longer side) Proof of Pythagoras’ Thereom ‘Both diagrams are of the same size square of side a + b. Both squares contain the same four identical right-angled triangles in white with sides a, b, c.

The left square also has two blue squares with areas a2 and b2 whereas the right hand one replaces them with one red square of area c2. This does not depend on the lengths a, b, c; only that they are the sides of a right-angled triangle. So the two blue squares are equal in area to the red square, for any right-angled triangle: a2 + b2 = c2 this makes an effective visual aid by pushing the squares from their locations on the left to where they are shown on the right. ‘ Prediction I predict that when I do this investigation I will find the rules and patterns of Pythagorean triples or triad.