To conduct Sports Day activities,in your rectangular shaped school ground $ABCD,$ lines have been drawn with chalk powder at a distance of $1 \,m$ each. $100$ flower pots have been placed at a distance of $1 \,m$ from each other along $AD,$ as shown in the figure. Niharika runs $\frac{1}{4}$ th the distance $AD$ on the $2^{nd}$ line and posts a green flag. Preet runs $\frac{1}{5}$ th the distance $AD$ on the $8^{th}$ line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags,where should she post her flag?

(N/A) It can be observed that Niharika posted the green flag at $\frac{1}{4}$ of the distance $AD,$ i.e.,$\left(\frac{1}{4} \times 100\right) \, m = 25 \, m$ from the starting point of the $2^{nd}$ line. Therefore,the coordinates of this point $G$ are $(2, 25).$
Preet posted the red flag at $\frac{1}{5}$ of the distance $AD,$ i.e.,$\left(\frac{1}{5} \times 100\right) \, m = 20 \, m$ from the starting point of the $8^{th}$ line. Therefore,the coordinates of this point $R$ are $(8, 20).$
Using the distance formula,the distance between these flags is $GR = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$
$GR = \sqrt{(8 - 2)^2 + (20 - 25)^2} = \sqrt{6^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61} \, m.$
The point at which Rashmi should post her blue flag is the mid-point of the line segment joining these points. Let this point be $M(x, y).$
$x = \frac{2 + 8}{2} = \frac{10}{2} = 5, \quad y = \frac{25 + 20}{2} = \frac{45}{2} = 22.5.$
Hence,the coordinates of the mid-point are $(5, 22.5).$
Therefore,Rashmi should post her blue flag at $22.5 \, m$ on the $5^{th}$ line.