Pythagorean Theorem Algebra Proof
What is the Pythagorean Theorem?
You can learn all about the Pythagorean Theorem, but here is a quick summary:
The Pythagorean Theorem states that, in a right triangle, the square of a (a^{2}) plus the square of b (b^{2}) is equal to the square of c (c^{2}):
a^{2} + b^{2} = c^{2}
Proof of the Pythagorean Theorem using Algebra
We can show that a^{2} + b^{2} = c^{2} using Algebra
Take a look at this diagram ... it has that "abc" triangle in it (four of them actually):
Area of Whole Square
It is a big square, with each side having a length of a+b, so the total area is:
A = (a+b)(a+b)
Area of The Pieces
Now let's add up the areas of all the smaller pieces:
First, the smaller (tilted) square has an area of | A = c^{2} | |
And there are four triangles, each one has an area of | A =½ab | |
So all four of them combined is | A = 4(½ab) = 2ab | |
So, adding up the tilted square and the 4 triangles gives: | A = c^{2}+2ab |
Both Areas Must Be Equal
The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:
(a+b)(a+b) = c^{2}+2ab
NOW, let us rearrange this to see if we can get the pythagoras theorem:
Start with: | (a+b)(a+b) | = | c^{2} + 2ab | |
Expand (a+b)(a+b): | a^{2} + 2ab + b^{2} | = | c^{2} + 2ab | |
Subtract "2ab" from both sides: | a^{2} + b^{2} | = | c^{2} |
DONE!
Now we can see why the Pythagorean Theorem works ... and it is actually a proof of the Pythagorean Theorem.
This proof came from China over 2000 years ago!
There are many more proofs of the Pythagorean theorem, but this one works nicely.