The Proof of Pythagoras’ Theorem The aim of the proof is to show that Pythagoras’ theorem is true for all right-angled triangles. The triangle shown above could be any right-angled triangle because its lengths are unspecified, and represented by the letters x, y and z. Also above, four identical right-angled triangles are combined with one tilted square to build a large square. It is the area of this large square which is the key to the proof. The area of the large square can be calculated in two ways. Method 1: Measure the area of the large square as a whole. The length of each side is x + y. Therefore, the area of the large square = (x + y)2. Method 2: Measure the area of each element of the large square. The area of each triangle is 1⁄2xy, i.e.1⁄2 × base × height. The area of the tilted square is z2. Therefore, area of large square = 4 × (area of each triangle) + area of tilted square Methods 1 and 2 give two different expressions. However, these two expressions must be equivalent because they represent the same area.
What do You think about Fermat's Last Theorem (1977)?
