Pythagorean Theorem Algebra Proof
What is the Pythagorean Theorem?
We have a page that talks all about the Pythagorean Theorem, but here is a quick summary:
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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
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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 |
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A = c² |
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| And there are four triangles, each one has an area of |
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A =½ab |
| So all four of them combined is |
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A = 4(½ab) = 2ab |
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| So, adding up the tilted square and the 4 triangles gives: |
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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: |
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(a+b)(a+b) |
= |
c² + 2ab |
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| Expand (a+b)(a+b): |
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a² + 2ab + b² |
= |
c² + 2ab |
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| Subtract "2ab" from both sides: |
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a² + b² |
= |
c² |
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DONE! |
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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.
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