The Pythagoras theorem is a fundamental principle in geometry that applies to right-angled triangles. It states that the square of the length of the hypotenuse, which is the side opposite to the right angle, is equal to the sum of the squares of the other two sides. If we call the two sides a and b, and the hypotenuse c, then the theorem can be written as a squared plus b squared equals c squared. This relationship is visually represented by squares drawn on each side of the triangle, where the areas of the squares on the two shorter sides sum to equal the area of the square on the hypotenuse.
One elegant way to prove the Pythagoras theorem is by comparing areas. Consider a large square with side length a plus b. We can divide this square in two different ways. First, we can see it as four right triangles, each with sides a and b, plus a smaller square in the middle with side length c, which is the hypotenuse of our triangles. Alternatively, we can view the same large square as consisting of two squares with areas a squared and b squared, plus four right triangles. Since both arrangements fill the same large square, they must have equal areas. This gives us the equation a squared plus b squared equals c squared, which is the Pythagoras theorem.