Figure 1

Here is one of the shortest proofs
of the Pythagorean Theorem.
Suppose we are given any right triangle
with sides of lengths A, B, C.
In order to show that
A^{2} + B^{2} = C^{2}
it is enough to show for any set of three
similar figures whose widths relate to each other
in the proportions A:B:C,
that the area of the largest figure is the
sum of the two smaller figures. This follows because
area scales like the square of the width. The figures
can be any shape.
So, let's use the original triangle for the shape of the
figures that we'll use!
Figure 1 shows that we've placed 3 blue triangles along
the sides of the ABC right triangle so that their
widths clearly relate to each other in the
proportions A:B:C.
The two smaller blue triangles are congruent
to the light and dark green right triangles,
and the largest blue triangle is congruent to the
entire ABC triangle. Since the sum of the areas of
the green triangles equals the area of the ABC triangle,
the sum of the areas of the smaller blue triangles
equals the area of the largest blue triangle,
as was to be shown!
Presentation Suggestions:
This is a lot easier to say than to write out in a
Fun Fact file! It's almost a proof without words!
The Math Behind the Fact:
Clever dissections of geometrical figures can give
new insights! It is fun to try to think of new
proofs to old theorems!
How to Cite this Page:
Su, Francis E., et al. "Shortest Pythagorean Proof?."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
