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 can be expressed as a squared plus b squared equals c squared.
To prove the Pythagorean theorem geometrically, we start with a large square that has side length a plus b. Inside this square, we arrange four identical right triangles. Notice how this arrangement creates a smaller square in the center with side length c, which is the hypotenuse of our triangles.
Now we calculate the areas of each component. The large square has area a plus b squared. Each triangle has area one half times a times b. Since we have four triangles, their total area is two a b. The center square has area c squared. Therefore, the total area equals two a b plus c squared.
Now we complete the algebraic proof. Starting with our area equation, we expand the left side to get a squared plus two a b plus b squared equals two a b plus c squared. Next, we subtract two a b from both sides, which gives us a squared plus b squared equals c squared. This is exactly the Pythagorean theorem, completing our proof!
In conclusion, we have successfully proven the Pythagorean theorem using geometric rearrangement. This elegant proof demonstrates how algebra and geometry work together to reveal fundamental mathematical truths. The Pythagorean theorem remains one of the most important and widely used theorems in mathematics, with countless applications in science, engineering, and everyday problem solving.