The Pythagorean theorem can be proven in many different ways. In this article we will show you one of these proofs of Pythagoras.
The theorem states that in a right triangle the square on the hypotenuse equals to the sum of the squares on the two legs.
Proof of the Pythagorean theorem
This proof is a bit like a puzzle where we use a square with a lot of right triangles.
The area $A$ of the blue big square is
$ A = (a+b)·(a+b) $
We can also express the area $A$ by using the small pieces (the smaller square and the four right triangles)
$A= c^2+4·\frac{a·b}{2} $
(The smaller square and the four right triangles that all has the same area)
Let´s simplify this by dividing $ \frac42 = 2 $
$A= c^2+4·\frac{a·b}{2} = c^2 + 2ab $
The both areas has to be equal so we get
$(a+b)·(a+b) = c^2 + 2ab$
Expanding $ (a+b)·(a+b)=a^2+2ab+b^2 $
$a^2+2ab+b^2 = c^2+2ab$
Subtract both sides with $2ab$
$a^2+b^2 = c^2$
And we are done with the proof! By using this geometric “puzzle” we have proven that the square $c$ equals the sum of the square of the two legs $a$ and $b$.