The Pythagorean Theorem

Perhaps the most famous theorem in mathematics is the Pythagorean Theorem. Most people who have had a high school geometry course still remember the statement   a2 + b2 = c2   long after their children (and even grandchildren) have learned it. The statement applies to a right triangle whose leg lengths are a and b, and whose hypotenuse has length c.

right triangle
a2 + b2 = c2

The name of the theorem comes from the ancient Greek mathematician and mystic, Pythagoras, but the theorem was known by the Babylonians before that. The earliest known proof, however, is Greek Since then, literally hundreds of proofs have been devised. Below is a picture proof found by James Garfield, twentieth president of the United States. Can you find the equations suggested by the figure which prove the theorem?

garfield's proof
A Presidential Proof

Here is another picture proof. First you might like to try to assemble the following "tangram" puzzle. The three pieces are shown on a grid so that you can cut accurate copies out of graph paper. When you assemble them, you may rotate and flip them, but you may not cut, fold, or overlap them.

piece 1 piece 2

piece 3

After you have tried the puzzle, read on. Pieces 2 and 3 form congruent right triangles: label their legs a and b, and their hypotenuses c. The single square made from the three pieces has side c, so its area is c2. The two squares made from the same pieces have sides a and b, respectively. So their areas are a2 and b2, respectively. Since the areas of the two figures must be the same, you have just proved that   a2 + b2 = c2 ! Continue on to see an animated solution:

Solution


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