Visual Representation of (a+b)(a-b)Date: 03/22/2003 at 15:03:12 From: Amber Subject: Is there a visual representation of (a+b)(a-b)? Is there a way to model (a+b) * (a-b)? I can use a rectangle array to model (a+b) times (a+b) but haven't been able to model a+b times a-b. Date: 03/22/2003 at 15:34:18 From: Doctor Rick Subject: Re: Is there a visual representation of (a+b)(a-b)? Hi, Amber. You can start with this figure: | a | - +-----------------+-------+ |/////////////////| | |/////////////////| | |/////////////////| | |/////////////////| | |/////////////////| | a |/////////////////| | |/////////////////| | |/////////////////| | +-----------------+-------+ |/////////////////| b | |/////////////////|b | |/////////////////| | - +-----------------+-------+ |/////////////////| b |/////////////////| |/////////////////| - +-----------------+ a-b The shaded area is (a+b)(a-b). Now compare it to this: | a | - +-----------------+-------+ |/////////////////|///////| |/////////////////|///////| |/////////////////|///////| |/////////////////|///////| |/////////////////|///////| a |/////////////////|///////| |/////////////////|///////| |/////////////////|///////| +-----------------+-------+ |/////////////////| b | |/////////////////|b | |/////////////////| | - +-----------------+-------+ This area is a^2 - b^2: the area of the big square, with the area of the little (b by b) square removed. Compare the two: they are equal, because the rectangle at the right of the second figure is congruent to the rectangle at the bottom of the first figure (it's just rotated 90 degrees). The remainder of the shaded region is the same in both figures. If I could work with colors or more subtle shading, I could have combined the two figures into one, so the relation would be more obvious. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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