Regions, Chords, and Circles

Date: 02/05/98 at 18:54:47
From: Dana Randolph
Subject: Geometry (regions, chords, and circles)

A given circle has n chords. Each chord crosses every other chord but 
no three meet at the same point. How many individual regions are in 
the circle?

This is how much I've completed on this problem:  I know that the xth 
line splits x regions, increasing the number of sections by x. I'm 
having trouble finding out the formula to solve the problem.
Please help me!

Date: 02/06/98 at 08:41:04
From: Doctor Jaffee
Subject: Re: Geometry (regions, chords, and circles)

Hi Dana,

Your observation about how the number of regions increases with each 
additional chord is right on target. We can use that to construct a 
table of values that can be helpful:

number of chords        1   2   3   4   5   6
number of regions       2   4   7  11  16  22

You noticed that the number of regions increases by 2, then 3, then 4, 
etc., and that is one of the properties of quadratic functions; that 
is, as the x number increases by 1, the y number increases by a 
constantly increasing amount. In the example above each increase is 
1 more than the previous increase.

So, we can pick any three ordered pairs from the chart, substitute the 
values into the standard equation of a quadratic function 
(y = ax^2 + bx + c). We will then have three equations in three 
variables which we can solve. Substitute those numbers back into the 
standard equation and we'll have finished.

It looks like this:

I would pick the first three pairs (1,2), (2,4), (3,7).
Substitute them into the standard quadratic form and get

        2 =  a +  b + c
        4 = 4a + 2b + c
        7 = 9a + 3b + c

The solution to this system is a = 1/2,  b = 1/2,  and c = 1.

So, substituting them into y = ax^2 + bx + c we get

   y = (1/2)x^2 + (1/2)x + 1

where x is the number of chords and y is the number of regions formed.  
You can then verify that if x = 4, y turns out to be 11 just as in the 
chart. If x = 5, y = 16, and so on.

I hope this has been helpful.  I enjoyed working on this problem.

-Doctor Jaffee,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/