Diagram to Prove Difference of Squares Formula

Date: 06/18/2008 at 18:07:50
From: Dina
Subject: link algebra to geometry 

Produce a diagram for the identity (a^2 - b^2) = (a + b) (a - b).

I have got (a + b)^2 = a^2 + b^2 + 2ab.  I found it easy to produce a 
diagram, link it to geometry to prove each side of the equation 
are equal. 

The area for the first square is a^2.
The area for the first rectangle is ab.
The area for the second square is b^2.
The area for the second rectangle is ab.
If we add them all together a^2 + ab + b^2 + ab = a^2 + 2ab + b^2.
                                       
But I do not know how to do (a^2 - b^2) = (a + b) (a - b).  Can you help?

Date: 06/18/2008 at 19:12:57
From: Doctor Achilles
Subject: Re: link algebra to geometry

Hi Dina,

Thanks for writing to Dr. Math.

Try drawing a square of sides a:

         a
    -----------
    |         |
    |         |
  a |         |
    |         |
    |         |
    -----------

The area of this is a^2.

Then, cut out a square of sides b:

         a
    -----------
    |         |
    |         |
  a |    -----|
    |    |    | b
    |    |    |
    -----------
          b

The area of the smaller square is b^2, so the area of the L-shaped
region is a^2 - b^2.

Now, draw an imaginary line through the L-shaped region:

        a
   -----------
   |         |
   |         |
 a |xxxx-----|
   |    |    | b
   |    |    |
   -----------
          b

Question 1: what is the area of the rectangle defined by the upper
part of this region?

Question 2: what is the area of the rectangle defined by the
lower-left part of this region?

Let me know what you get.  If you need more help linking this idea
back to the algebraic expression (a+b)(a-b), please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/