The Pythagorean theorem is one of the most famous theorems in mathematics. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This relationship is expressed as a squared plus b squared equals c squared.
To prove the Pythagorean theorem, we start by constructing a large square with side length a plus b. Inside this square, we arrange four identical copies of our right triangle. Notice how the hypotenuses of these triangles form a smaller square in the center.
Now let's calculate the areas. The large square has area a plus b squared. Each right triangle has area one half a b, so four triangles have total area two a b. The inner square formed by the hypotenuses has area c squared. Therefore, the area of the large square equals the sum of the triangle areas plus the inner square area.
Now we complete the algebraic proof. Starting with our equation a plus b squared equals two a b plus c squared, we expand the left side to get a squared plus two a b plus b squared equals two a b plus c squared. Finally, we subtract two a b from both sides to obtain a squared plus b squared equals c squared, which is the Pythagorean theorem.
To summarize what we have learned: The Pythagorean theorem establishes a fundamental relationship between the sides of right triangles. We proved it using geometric area relationships by constructing squares and triangles. The formula a squared plus b squared equals c squared applies universally to all right triangles and remains one of the most important theorems in mathematics.