geometry question

Question

 A plane contains 11 horizontal lines and 11 vertical lines. These lines divide the plane into disjoint regions. How many of these disjoint regions have a finite nonzero area.

 

A) 100

B)110

C)144

D)156

E)169

 

Any help will be appreciated. Please try to show all work. THANK YOU!!!!

 

 

Y(10)

Answer 1

A lot of times with problems like these, it's easier to picture what's going on if you start with smaller numbers. So, think about what happens when there are two horizontal and two vertical lines. You could easily draw this situation out and see that your four lines will create a single finite non-zero region. The other eight regions are infinite.

 

What happens if there are three horizontal lines and three vertical lines? Again, you can draw this out. You'll see that the lines create four finite non-zero regions.

 

For four of each, the lines create nine finite non-zero regions.

 

This happens because each case where there are n lines results in an enclosed grid of length (n-1)×(n-1). So, with 2 lines, you get a 1×1 grid (with 1×1 = 1 square). With 3 lines, you get a 2×2 grid (with 2×2 = 4 squares). This pattern continues.

 

 

 

Applying that to your situation, you can tell that 11 lines will produce a 10×10 grid. That means you'll have 10×10 = 100 finite non-zero-area regions.

Answer 2

Lines have infinite length. For simplicity sake let us assume the lines are equally spaced creating a large square. The perimeter of the square is created by the most left and right vertical lines and the top and bottom most horizontal lines. 

 

Now the most left line has infinite regions to the left of it, but the other 10 lines have finite regions to the left of them.

 

We can think of the horizontal lines the same way. The bottom most line has infinite regions below it, but the other 10 lines have finite regions below them.

 

This makes a 10 by 10 finite grid so there are 100 finite regions created by the 22 intersecting lines.