Re: factoring using geomerty?

pleyland_at_microsoft.com
Date: 09/29/03 

Date: 29 Sep 2003 12:24:57 +0100

"sqrt\(i\)" <com.com@com> writes:

> would you care to elaborate just a little on the link between the difference
> of squares and its relation to geometry?

Bob can undoubtedly answer for himself, but this example may be
useful.

Draw two squares, one of side x and another with side y. Assume x>y
and that the smaller square shares two edges and one corner with the
larger. Something like this:

  +--+-----+
  | | |
  | | |
  | | |
  | | |
  | | |
  ####-----+
  #### |
  #### |
  ####-----+

which is a 4x4 square (so y = 4) inside a 10x10 square (and x=10).

I've drawn in two other lines. The difference between the two squares
is the unshaded area. The two portions abutting the smaller square
are clearly rectangles with sides y and (x-y). The remaining
unshading area is clearly a square with sides (x-y). Now take one of
the rectangles and place it alongside the square opposite the other
rectangle. You get a long thin rectangle. One side is clearly still
just (x-y). The other side is (x + y) in length. The area of this
new rectangle is thus (x-y)*(x+y) but, by construction, the area is
the same as the difference in area of the two squares. Conclude that
x^2-y^2 = (z-y)*(x+y).

The diagram after rearrangement looks something like this:

  +--+-----+--+
  | | | |
  | | | |
  | | | |
  | | | |
  | | | |
  ####-----+--+
  ####
  ####
  ####

Paul 

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