The Pythagorean theorem is one of the most famous results 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. Today we will explore a beautiful visual proof that demonstrates this relationship through geometric areas.
Now we begin our visual proof. First, we create 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 with side length c.
Now we create a second arrangement using the same large square with side length a plus b. This time, we rearrange the four triangles differently. Two triangles form one rectangle, and the other two form another rectangle. This arrangement leaves two squares: one with side a and area a squared, and another with side b and area b squared.
Now comes the key insight of our proof. Both square arrangements have exactly the same total area, which is a plus b squared. The first arrangement gives us an area of two a b plus c squared. The second arrangement gives us an area of two a b plus a squared plus b squared. Since both expressions equal the same total area, they must be equal to each other.
To summarize what we have learned: This visual proof demonstrates the Pythagorean theorem by comparing areas of geometric shapes. By arranging the same triangles in two different ways within identical squares, we revealed the fundamental relationship. The algebraic manipulation of equal areas leads directly to the famous equation c squared equals a squared plus b squared.