Circle divided??

Question

When a circle is divided into regions by drawing one straight line through it the number of regions is two. If two lines are drawn through the circle there can be three or four regions resulting. How many regions are possible if three lines are drawn through the circle? What if fours lines are drawn? Can you show the work? 

 

THANK YOU FOR YOUR HELP!!

Answer 1

You have to do the drawings!!

 

Hopefully, the first example is obvious---one line divides any plane figure into 2 parts.

 

In the second example, the key question is whether the 2 lines cross inside the circle.  If so, then we get 4 sections.  If not, we get 3.

(Draw the pictures so you see this.)

 

Continuing---the number of sections that are created will always depend on how many line crossings there are.

 

You can easily see that the maximum number of sections occurs if every line is drawn so that it maximizes the number of crossings.

 

A few minutes of drawing will allow you to see the pattern and the range of possibilities