The Many Proofs of the Pythagorean Theorem

          There are about 45 proofs for the Pythagorean Theorem. We will learn the most famous of those in the Proofs section.
These proofs will require some vivid visualization. Let's get started, shall we?

The Rearrangement Proof

          The rearrangement proof starts off by creating three more copies of the same triangle. Next, we have to put them
together so that both legs (Side A & Side B) are on the outside each time. As you can see in the picture below, they form a
square with an area of the Hypotenuse squared (C²). With this, you can tell that A² + B² = C².

Example 3.1

Prove that the hypotenuse of this triangle of the above triangle is 13.
First, we have to take four of this triangle and stick them together, like this:

Now let's solve for A² + B². 5² is 25. 12² is 144.
Now we have to add 144 and 25. We get 169.
Next, we need to find the area of the white square in the center of the picture.
Since 13 is the Hypotenuse (C), we need to do 13 x 13. We get 169.
169 = 169, so the Pythagorean theorem is correct!

The Rectangular Rearrangement Proof

          The Rectangular Rearrangement Proof follows the same basic principles as the Rearrangement proof stated above.
In the Rectangular Rearrangement Proof, we rearrange four of the triangles so that they form two rectangles. Once we
rearrange it it should look somewhat like this:


As you can see, the two white squares have the area of A² and B² respectively.

Example 3.2

Prove that the hypotenuse of this triangle of the above triangle is 5.
First, we have to take four of this triangle and stick them together, like this:

Now let's solve for A² + B². 3² is 9. 4² is 16.
Now we have to add 9 and 16. We get 25.
Next, we need to take the square root of 25, it is 5.
Abracadabra, 5 = 5, so the Pythagorean Theorem has proved right once again!