The Doubling Proof 
Still not convinced, that the Pythagorean theorem is a true fact? Let's learn the Doubling Proof.
The Doubling Proof appears very hard, but is actually very simple. First, look at the picture below.
To find the area of the triangle ABD, we have to solve this equation:
AD x DB x 1/2
If we take away the "x 1/2" then it would be the same as doubling the equation and making the
triangle into a rectangle. Let's go ahead and do that to triangle ABD and triangle ADC.
Now, triangle ABC has turned into this shape after doubling triangles ABD and ADC and turning
and shearing the shapes.
This rectangle has an width of AB, and a length of AC.
Therefore, the area of this rectangle is BCA2 (AB x AC).
The Pythagorean Theorem has proved yet again in being correct!
The Squaring Proof
The squaring proof is the last proof we will visit today. It starts
off
with the same, simple right triangle. First, we have to square all of
the sides, like this:
We then calculate the area of all of the squares. Next, we add up
the area of the two smaller squares.
We get the area of the large
square. Therefore, the Pythagorean Theorem is completely true.
Example 3.3
Prove that the hypotenuse of this triangle of the above triangle is 5.
First, we have to make squares of all of the sides, like this:
The area of the smaller squares are 9 and 16 respectively.
9 + 16 = 25, which is the area of the large square,
So the Pythagorean Theorem is right!
The Total Number of Proofs
One would think that many people did not believe Pythagoras when he came up with this formula.
By the way, you might want to visit the Pythagoras page to learn about him. There are an astounding 44
proofs for this theorem! We have only visited 4 of them, if you would like to learn them all you might want
to visit the Pythagorean Theorem and it Many Proofs.
