Pythagorean Theorem

The Pythagorean theorem helps us find an unknown side of a right triangle. The sides that form the right angle are the legs. The longest side, opposite the right angle, is the hypotenuse.

The theorem states that the sum of the squares of the legs equals the square of the hypotenuse.

Let's prove it. Imagine four identical right triangles arranged inside a large square. Each triangle has legs a and b, so the large square has side length a plus b and area a plus b, squared.

Inside it, the triangles surround a smaller square with side length c, so its area is c squared. The four triangles have a total area of 2ab. So, the large square’s area is also c squared plus 2ab.

Since both expressions give the same area, a plus b, squared equals c squared plus 2ab. Expanding and simplifying this equation gives the Pythagorean theorem.

Let's see an example. A carpenter is building a gate that is 4 meters tall and 3 meters wide. He needs to find the diagonal brace length.

Since the height, width, and brace form a right triangle, the theorem applies. Substituting the values and taking the square root shows that he needs a 5-meter brace.