To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs $\dfrac{1}{4}$ th the distance AD on the 2nd line and posts a green flag. Preet runs $\dfrac{1}{5}$ th the distance AD on the eighth 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?

Given,

100 flower pots have been placed at a distance of 1 m from each other along AD.

AD = 100 × 1 = 100 m.

It can be observed that Niharika posted the green flag at $\dfrac{1}{4}$th of the distance AD i.e., $\dfrac{1}{4} \times 100 = 25$ m from the starting point of 2nd line. Therefore, the coordinates of this point G is (2, 25).

Similarly, Preet posted red flag at $\dfrac{1}{5}$th of the distance AD i.e.,$\dfrac{1}{5} \times 100 = 20$ m from the starting point of 8th line. Therefore, the coordinates of this point R are (8, 20).

By formula,

Distance between two points = $\sqrt{(y$2 - y1)^2 + (x2 - x1)^2}

Substituting values we get :

$GR = \sqrt{(20 - 25)^2 + (8 - 2)^2} \\[1em] = \sqrt{(-5)^2 + (6)^2} \\[1em] = \sqrt{25 + 36} \\[1em] = \sqrt{61} \text{ m}.$

The point at which Rashmi should post her blue flag is the mid-point of the line joining these points. Let this point be A (x, y).

By formula,

Mid-point = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get :

$A (x, y) = \Big(\dfrac{2 + 8}{2}, \dfrac{20 + 25}{2}\Big) \\[1em] = \Big(\dfrac{10}{2}, \dfrac{45}{2}\Big) \\[1em] = (5, 22.5)$

From figure,

The x-coordinate represets the no. of line and y-coordinate represents the vertical distance.

Hence, Rashmi should post her blue flag at a distance of 22.5m on 5th line.