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 and a and b are the legs of the triangle.
This proof arranges four identical right triangles within a large square. The area of the large square equals a plus b squared. This area also equals the sum of four triangle areas plus the inner square area c squared. By algebraic manipulation, we get a squared plus b squared equals c squared.
Pythagorean triples are integer solutions to the theorem. The most well-known is three, four, five, where nine plus sixteen equals twenty-five. Other common triples include five, twelve, thirteen and eight, fifteen, seventeen. These triples have practical applications in construction and navigation.
The Pythagorean theorem appears everywhere in real life. Construction workers use it to ensure buildings are square and to calculate ladder placement. In navigation, it helps determine the shortest distance between two points. Engineers apply it in structural design, and GPS systems use it for positioning calculations.