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
All three triangles have equal angles and are
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.