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# 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 given figure below. Niharika runs $\frac{1}{4}$th the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$th 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:

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. Niharika runs $\frac{1}{4}$th the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$th distance $AD$ on the eighth line and posts a red flag.

To do:

We have to find the distance between both the flags and if Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, the distance at which she should post her flag.

Solution:

y-coordinate of the green flag $=\frac{1}{4}\times100$

$= 25$

This implies,

Coordinates of the green flag are $P (2, 25)$

y-coordinate of the red flag $= \frac{1}{5}\times100$

$= 20$

This implies,

Coordinates of red flag are $Q (8, 20)$

Using the division formula

$(x,y)=[\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}]$

The distance between two points is

$\mathrm{PQ}=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$\mathrm{PQ}=\sqrt{(8-2)^{2}+(20-25)^{2}}$

$=\sqrt{36+25}$

$=\sqrt{61} \mathrm{~m}$

Position of the blue flag $=$ The mid-point of $\mathrm{PQ}$

$=\frac{2+8}{2}, \frac{25+20}{2}$

$=(5, 22.5)$

The blue flag is in the 5th line at a distance of $22.5\ m$.

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