Help with a math kangaroo question

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

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 \( 3 \).