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. We write this as a squared plus b squared equals c squared, where a and b are the legs of the triangle, and c is the hypotenuse, which is always the longest side opposite the right angle.
To visualize the Pythagorean theorem, we construct squares on each side of our right triangle. The square on side a has area a squared, the square on side b has area b squared, and the square on the hypotenuse c has area c squared. This visual representation clearly shows the relationship: the sum of the areas of the two smaller squares equals the area of the largest square, which is exactly what the Pythagorean theorem states.
Now we set up our geometric proof. We start with a large square having side length a plus b. Inside this square, we can arrange four identical right triangles in different ways. In the first arrangement, the triangles leave two separate squares: one with area a squared and another with area b squared. We will see how this leads us to prove the Pythagorean theorem through area comparison.
Now we execute the main proof by calculating areas. The total area of the large square is a plus b squared. In the first arrangement, this equals four triangles each with area one-half a b, plus the two squares with areas a squared and b squared. In the second arrangement, it equals the same four triangles plus one square with area c squared. Since both expressions equal the total area, we can set them equal and simplify to get a squared plus b squared equals c squared, proving the Pythagorean theorem.
Let's reinforce our understanding with a concrete numerical example. We have a right triangle with legs of length 3 and 4 units. Applying the Pythagorean theorem: 3 squared plus 4 squared equals c squared. This gives us 9 plus 16 equals c squared, so 25 equals c squared, therefore c equals 5. We can verify this by showing the squares with areas 9, 16, and 25, confirming that 9 plus 16 indeed equals 25. This creates the famous 3-4-5 right triangle.