The Pythagorean Theorem and Its Geometric Proof

The Pythagorean Theorem is one of the fundamental principles in mathematics, defining the relationship between the sides of a right triangle. This theorem states that in a right 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 other two sides. In mathematical terms, for a triangle with sides a, b, and c (where c is the hypotenuse), the theorem can be expressed as:

Proof: Geometric Approach

The geometric proof of the Pythagorean Theorem is both elegant and insightful. Consider four triangles with sides a and b, forming squares on their respective hypotenuses. Arrange these squares to form a larger square, where the outer square has a side length of a+b. The area of the outer square is (a+b)².

The sum of the areas of the four triangles and the central square equals the area of the larger square with a side length of (a+b). When we substitute this into the equation, we obtain the following:

Essentially, it shows that the square of the hypotenuse’s length is equal to the sum of the squares of the other two sides.