There are many, many different proofs of this famous theorem. An internet search for "Pythagorean Theorem" + proof yields over 110,000 references. There are geometric proofs, algebraic proofs, and a variety of other types of proofs. The proof in this project is visual and "hands-on", while at the same time, getting right to the heart of the matter. When students see this, they will gain an intuitive understanding of the theorem, an understanding that will help them to remember not only the theorem, but it's meaning. When they construct it themselves, they gain an even deeper, concrete understanding of the theorem.
Constucting the "Visual Proof":
1) Draw or construct a square. Label each side "c" as shown.
2) Construct a short segment on the top line of the square. Construct 3 more segments equal in length to the first, as shown below. Label each small segment "b" and each longer segment "a" as shown:
3) Join the 4 points as shown to create a second, smaller square. Label each side of the smaller square "c" as shown. Notice that you now have 4 triangles inside and a square insiden the original square:
4) Fold the four outer triangles in towards the center of the square as shown below. Notice that the lettering for the new position of each point (after folding) is not bold, so you can tell one from the other. Also, the previous position of each folded portion is a dashed line. Hopefully, this makes the drawing easier to understand. Also, remember from step one that each entire side of the original square is length "c". You now have completed the paper model.
Now, here comes the math! The area of each triangle is 1/2 base times height, which is equal to:
There are 4 triangles, so the area of all 4 of them is 4(1/2ab) = 2ab.
The area of the largest square is (a+b)(a+b),which is equal to:
So, if we subtract the area of the 4 yellow triangles we will get ;
Which is . . .
The Pythagorean Theorem!
Plato
(ca 429-347 BC)
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