Welcome to our exploration of the Pythagorean theorem. This fundamental theorem in geometry states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. We can express this as a squared plus b squared equals c squared, where c is the hypotenuse and a and b are the legs of the triangle.
Now we begin our proof by constructing a large square with side length a plus b. The area of this large square is a plus b squared, which when expanded gives us a squared plus two a b plus b squared. We divide this square into sections using internal lines to help visualize our proof.
Next, we place four identical right triangles in the corners of our large square. Each triangle has legs of length a and b, and hypotenuse of length c. The area of each triangle is one half times a times b. Since we have four triangles, the total area is four times one half a b, which equals two a b. Notice that the hypotenuses of these triangles form a smaller square in the center with area c squared.
Now we set up our key equation. The area of the large square equals the sum of the areas of the four triangles plus the area of the inner square. This gives us the equation: a plus b squared equals two a b plus c squared. Expanding the left side, we get a squared plus two a b plus b squared equals two a b plus c squared. Subtracting two a b from both sides, we arrive at a squared plus b squared equals c squared, which is the Pythagorean theorem.
We have successfully derived the Pythagorean theorem using a geometric proof. By constructing a large square and arranging four right triangles within it, we showed that the area relationships lead directly to the famous equation a squared plus b squared equals c squared. This elegant proof demonstrates the fundamental relationship between the sides of a right triangle and has been used for over two thousand years in mathematics and engineering.