## The Pythagorean Theorem

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 Pythagorean Theorem**
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 .

Let

denote the area of the large square. It can also be
computed by adding the areas of the four triangles and the area of the
inner square. The area of one blue triangle equals
(which
is half of base times height). Thus we also have
The two expressions are equal and we
obtain: