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 $3$.