The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of 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.
This proof arranges four identical right triangles in a square pattern. The outer square has side length a plus b, so its area is a plus b squared. The inner tilted square represents the hypotenuse c, and the four triangles each have area one-half a b.
Now we solve algebraically. We expand a plus b squared to get a squared plus two a b plus b squared. This equals four times one-half a b plus c squared, which simplifies to two a b plus c squared. Subtracting two a b from both sides gives us a squared plus b squared equals c squared.
The Pythagorean theorem has countless practical applications. Consider this ladder problem: a five-meter ladder leans against a wall with its base three meters from the wall. To find the height, we use h squared plus three squared equals five squared. Solving gives us h squared equals sixteen, so h equals four meters.