Diagram to Prove Difference of Squares FormulaDate: 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/ |
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