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 (a2) plus the square of b (b2) is equal to the square of c (c2):
a2 + b2 = c2
Proof of the Pythagorean Theorem using Algebra
We can show that a2 + b2 = c2 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 = c2|
|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 = c2+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) = c2+2ab
NOW, let us rearrange this to see if we can get the pythagoras theorem:
|Start with:||(a+b)(a+b)||=||c2 + 2ab|
|Expand (a+b)(a+b):||a2 + 2ab + b2||=||c2 + 2ab|
|Subtract "2ab" from both sides:||a2 + b2||=||c2|
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.