Monday, November 29, 2010

Another Proof of Pythagoras Theorem

Pythagoras Theorem is one of the most important theorems in elementary geometry. It states that in a right angled triangle the square of the largest side is the sum of the squares of the other two sides.

There are a large number of proofs available in textbooks and online articles. I will show here a very simple proof that uses only two concepts:

1) Area is additive.
2) Congruent figures have same area.

The proof falls in the category of "Proof Without Words" (may need a line or two of explanation):

As can be seen both squares are of side (a + b) and contain 4 right triangles of equal area (in fact they are congruent triangles) in the shaded portion. It follows that the remaining area (in white) in both the squares is same and therefore we have a^2 + b^2 = c^2.

This is something of a marvel as you don't even need to know the formula of the area of a triangle. If you know the formula for the area of the triangle then the figure on the left is enough to prove the Pythagoras theorem:

Area of bigger square = (a + b)^2 = Area of 4 triangles + Area of smaller square (in white)
and so (a + b)^2 = 4 * (ab / 2) + c^2 and thus a^2 + b^2 = c^2

The proof above using two figures is from 7th standard NCERT textbook. And it again shows that NCERT beats any other textbook in the school level curriculum. Next time I will show another proof about sum of angles of a polygon.


Blogger tsg said...


I am trying to place the mathjax code on my blog

it is not working , can me help me out.

5:09 PM  

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