Prove Pythagoras' Theorem
Never in a GCSE syllabus, but still a question I had as a student and one I hear today- something, I feel, is worth going over (Similar in a sense to proving that completing the square works). We use the "dissection and rearrangement" proof. We take a square of side length (a+b), and put a tilted square inside that one of length c, so that the corner points of the little square touch the bigger square "a" away from one corner and "b" away from the other. A diagram will make this clear.
The area of the whole, bigger square is (a+b)(a+b) = a2 + b2 + 2ab (using FOIL expansion)The bigger square is made up 5 components: 4 right angled triangles and the little square. The area of the little square is c^2. The area of each of the four triangles are (ab)/2. So in total we have c^2 + (ab)/2 * 4.The areas of the bigger square and the 5 components must be equal. SO:a2 + b2 + 2ab = c 2 + 2ab gives a2 + b2 = c2
