The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This fundamental relationship is expressed as a squared plus b squared equals c squared, where c is the hypotenuse.
We can visualize the Pythagorean theorem by constructing squares on each side of the 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. The theorem states that the sum of the areas of the two smaller squares equals the area of the largest square.
Let's verify the Pythagorean theorem with a classic example: the 3-4-5 triangle. We calculate 3 squared equals 9, 4 squared equals 16, and 5 squared equals 25. Adding the squares of the legs: 9 plus 16 equals 25, which indeed equals 5 squared. This confirms that the 3-4-5 triangle is a perfect right triangle.
The Pythagorean theorem is useful for finding unknown sides of right triangles. Given a triangle with legs of length 6 and 8, we can find the hypotenuse. We substitute into the formula: 6 squared plus 8 squared equals c squared. This gives us 36 plus 64 equals c squared, so 100 equals c squared. Taking the square root, we find c equals 10.
The Pythagorean theorem has countless real-world applications. Architects use it to ensure buildings are square and stable. GPS systems rely on it for accurate positioning. Computer graphics use it for distance calculations and 3D rendering. Engineers apply it in structural design and physics calculations. This ancient theorem, discovered over 2500 years ago, remains one of the most practical and fundamental principles in mathematics today.