Proof of Pythagoras' Theorem

Pythagoras' Theorem is a cornerstone of geometry, stating that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides: a² + b² = c². While many might recall the formula, mathematics demands a "proof" to establish its universal truth. A proof demonstrates *why* a statement is always true, moving beyond mere observation or specific examples.

One of the most elegant and widely understood proofs utilizes area. Imagine taking four identical right-angled triangles, each with sides of length 'a' and 'b' and a hypotenuse of length 'c'. We can arrange these four triangles within a larger square.

Consider a large square whose side length is (a+b). Its total area would therefore be (a+b)². Now, arrange the four identical right-angled triangles inside this large square such that their hypotenuses form the boundaries of a smaller, central square. The four vertices of the inner square will be at the corners of the four triangles. The side length of this inner square will be 'c', making its area c².

The area of each of the four right-angled triangles is ½ * base * height, or ½ab. So, the combined area of the four triangles is 4 * (½ab) = 2ab.

The total area of the large square can thus be expressed in two ways: 1. As the square of its side: (a+b)² 2. As the sum of the areas of its constituent parts: the four triangles plus the inner square, which is 2ab + c².

Setting these two expressions for the area equal to each other: (a+b)² = 2ab + c²

Now, let's expand the left side of the equation: a² + 2ab + b² = 2ab + c²

Finally, subtract 2ab from both sides of the equation: a² + b² = c²

This algebraic simplification proves the theorem: the sum of the squares of the two shorter sides of a right triangle is indeed equal to the square of its hypotenuse. This simple yet profound demonstration reveals the theorem's inherent truth, applicable to every right-angled triangle, forever transforming it from an observation into an undeniable fact.