Geometrically Completing the SquareDate: 08/07/97 at 11:24:00 From: Katie Brennan Subject: Geometrically completing the square Hi, I've been searching everywhere to find the steps for geometrically completing the square. Do you know them? Katie Date: 08/07/97 at 11:58:03 From: Doctor Jerry Subject: Re: Geometrically completing the square Hi Katie, Start with a rectangle. Our job is to find a square with equal area. So that we have the same picture in our heads, please draw a rectangle on your paper. Make it 2 inches by 1 inch, with the 2 inches horizontal. Mark the vertical side, on the left, x. Mark off x on the horizontal side, starting on the left. Now you have a square with sides x. Draw the vertical line to make the square. On the top, put 7 on the remaining segment. It doesn't matter that it's 7. The rectangle has area 7x. Right? Okay, we have a rectangle whose area is x^2+7x. To complete the square algebraically, we write x^2+7x+(7/2)^2 - (7/2)^2 = (x+7/2)^2 - (7/2)^2. Geometrically, divide the thin rectangle on the right by a vertical line down the middle. Move the rightmost half down and to the left. If you rotate it 90 degrees, it will just fit under the square. Now you see a figure with area equal to the original rectangle, but arranged so that we can complete the square. What's missing is the little square down in the lower righthand corner. What is its area? Well, you will see that its area is (7/2)^2. So, returning to the original question, to find a square with area equal to the original rectangle all we must do is to start with two squares and construct their difference. This can be done by a geometrical procedure too. The two squares have sides x+7/2 and (7/2)^2. Neat, huh? All of this was known to the Vedic Indians (Asian Indians), quite a long time before Euclid. It's in their religious book, the Sulvasutra. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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