## Proof of the Pythagorean Theorem

There are many ways to prove the
Pythagorean Theorem. A particularly
simple one is to use the scaling relationship for
areas of similar figures.

Consider any right triangle ABC.
Choose point D on the hypotenuse AB, such that line CD
is perpendicular to AB. Then the large right triangle
ABC is split into two smaller right triangles ADC and
BDC.

All three triangles have equal angles and are
therefore

*similar*, so their
areas are related by the scaling formula:

for some number "s" that is the same for
all three triangles.

(Scroll down
to see the rest of the proof.)

The two small triangles exactly
cover the large triangle, so the large area must equal
the sum of the two smaller areas. Since "c", "a", and "b"
are the corresponding sides, we can write the equation:

But then simple algebra tells us that we can divide both
sides of the equation by the number "s" to eliminate
that symbol (even though we never know
what its value is!). This leaves us with
the Pythagorean Theorem:

So, the Pythagorean Theorem is true just because of a simple scaling
law:

**
For similar shapes, area increases as the ***square*
of the lengths of corresponding sides.