Geometrically Completing the Square

Date: 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/