Why is Pythagoras theorem (a^2 + b^2 = c^2) true for every right angle triangle?
Firstly let's take a right angle triangle. Label its sides a,b, c for the hypotenuse. Then arrange 4 of these triangles into a large square, with a smaller square contained within it. The area of the smaller square will all be c and so the size of the smaller square is c2 . Now the area of the larger square can be expressed in 2 ways firstly by taking its base and height (both a+b). Giving us the value for its area (a+b)2 , the second way to express the area of this square is to add up all the components of it (the four triangles and the smaller square). The area of an individual triangle is 1/2base x-height (its base and height being a and b) and the area of the small square as previously calculated is c2 so we add these together and we get 2ab+c2 for the area of the large square. Put our two values for its area together we end up with the equation (a+b)2 = 2ab + c2. (At this point the student could expand the brackets on (a+b)2). when you expand the brackets on (a+b)2 you get a2 + b2 +2ab = 2ab + c2 . Take 2ab off from both sides and you get a2 + b2 = c2 .
