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The smallest number of straight lies that will divide a plane into 5 regions is?

 Mar 2, 2024
 #1
avatar+129690 
+1

Formula  for the number of lines, n,  that will divide a plane into k regions

 

(1/2) (n^2 + n + 2)  =  k

 

(1/2) (n^2 + n + 2)  =   5

 

n^2 + n + 2  = 10

 

n^2 + n =  8

 

n  must be  greater than 2

 

cool cool cool

 Mar 3, 2024
 #3
avatar+2 
+1

To divide a plane into \( 5 \) regions using straight lines, we need to determine the minimum number of lines required to achieve this configuration.

Let's break down the problem:

 

1. With no lines, the plane forms one region.
2. With one line, the plane is divided into two regions.
3. With two lines, the plane can form up to \( 4 \) regions.
4. With three lines, the plane can form up to \( 7 \) regions.

 

Now, we need to find the minimum number of lines required to form \( 5 \) regions.

 

If we draw \( 3 \) lines, we will have \( 7 \) regions. But if we add a fourth line, it will intersect with the existing regions, increasing the number of regions by \( 1 \). Thus, with \( 4 \) lines, we can have \( 8 \) regions, which is more than required.

Therefore, the smallest number of straight lines that will divide a plane into \( 5 \) regions is \( 3 \).

 Mar 4, 2024

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