Here is a right triangle.
The Pythagorean theorem says that the square of the longest side,
called the hypotenuse, is the sum of the square of the two shorter
sides, called legs, or written in a formula,
c2=a2+b2
Every proof I know of this fact has to do with area, and the one I
am about to give is no exception. It is based on the proof that I
learned when I was in high school, but with the algebra taken out of
it, so that it can be understood by students with little or no algebra
background. The idea is that since the area of a square is the square
of its side length, we will have proved the theorem if we can show that
the area of a square of side length c is the sum of the areas of
squares of side lengths a and b. To make this more clear, here are
those squares, drawn attached to the triangle.
What we need to show is that the yellow area is the same as the
combined red and blue areas. One way we can do that is if we could show
how the blue and red squares could be cut up and put together in a
puzzle to make the yellow square. In order to do this what we
will do is to get four triangles like the original triangle and a
square of side length b-a out of the red and blue square. Then what we
can do is put each of the triangles on a side of square leaving a
little space in the middle where the square will fit. If that doesn't
quite sound clear, don't worry, it will become clearer when we do it.
To get those triangles and square, what we do is we cut the bottom
strip off of the blue square to make an a by b rectangle and connect
this strip to the red square to make another a by b rectangle with a
square with side length b-a left over like this.
Now we cut off the little square and cut across the diagonal of the two
rectangles to make the four triangles like this.
I've put the little square in the center, because that is where it
needs to go. Now all we have to do is fit in the triangles. To do that,
fit each of them around the square with their hypotenuses lined up with
the sides of the triangle. That part is left as an exercise for the
students. To do that, here is a picture to use for the actual cutting
and pasting. If you print this out and cut on the lines and follow the
above directions, you should be able to fit the red and blue pieces to
make the yellow square.
