The Pythagorean theorem is one of the most famous mathematical relationships. It states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This fundamental relationship, expressed as a squared plus b squared equals c squared, applies only to right triangles where c represents the hypotenuse, the longest side opposite the right angle.
The most intuitive proof of the Pythagorean theorem uses squares constructed on each side of the triangle. When we build squares on the legs with sides a and b, their areas are a squared and b squared respectively. The square on the hypotenuse has area c squared. For our 3-4-5 triangle example, the squares have areas 9, 16, and 25. Notice that 9 plus 16 equals 25, visually confirming that a squared plus b squared equals c squared.
The Pythagorean theorem can be rearranged algebraically to solve for any unknown side. Starting with a squared plus b squared equals c squared, we can solve for the hypotenuse by taking the square root: c equals the square root of a squared plus b squared. To find a leg, we rearrange to get a equals the square root of c squared minus b squared. For example, with legs of 3 and 4, the hypotenuse is the square root of 9 plus 16, which equals 5.
The Pythagorean theorem solves many real-world problems. First, finding a rectangle's diagonal: for a 6 by 8 rectangle, the diagonal equals the square root of 6 squared plus 8 squared, which is 10. Second, ladder safety: if a 13-foot ladder reaches 12 feet up a wall, the base must be 5 feet from the wall, calculated as the square root of 13 squared minus 12 squared.
Advanced applications include Pythagorean triples - integer solutions like 3-4-5, 5-12-13, and 8-15-17. The theorem extends to coordinate geometry through the distance formula: the distance between two points equals the square root of the sum of squared coordinate differences. For example, the distance from point (1,2) to (4,6) is 5 units, calculated as the square root of 3 squared plus 4 squared.