Some of the homework problems in this class require the Pythagorean Theorem, named after Pythagoras who lived approximately 380-300 BC. That Theorem is also used for example when computing the distance between two points in the Cartesian Coordinate System.
The Pythagorean Theorem states that in a right triangle the squares of the two short sides add to the square of the long side. If we call the lengths of the two short sides and , and the length of the long side this leads to the familiar statement
Of course, the sides need not be called , and , and the reverse of the above statement also holds: if the equation holds than the triangle in question is a right triangle.
The Figure on this page illustrates a simple proof of the Pythagorean Theorem. We describe that proof here because it provides a beautiful application of Intermediate Algebra. Take four right triangles (shown in blue) with sides , and and line them up as indicated in the Figure. Thus we continue the side of one triangle with the side of another. The result is a square whose sides have length . It encloses a smaller square (shown in red) whose sides have length .