To conduct Sports Day activities, in your rectangular shaped school ground ABCD,
Question.

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

Solution:

Let us consider ‘A’ as origin, then

AB is the x-axis.

Now, the position of green flag-post is

$\left(\mathbf{2}, \frac{\mathbf{1 0 0}}{\mathbf{4}}\right)$ or $(2,25)$

And, the position of red flag-post is

$\left(\mathbf{8} \frac{\mathbf{1 0 0}}{\mathbf{5}}\right)$ or $(8,20)$

$\Rightarrow$ Distance between both the flags

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

$=\sqrt{6^{2}+(-5)^{2}}=\sqrt{36+25}=\sqrt{61}$

Let the mid-point of the line segment joining the two flags be M(x, y).

$\therefore \quad x=\frac{2+8}{2}$ and $y=\frac{25+20}{2}$

or $x=5$ and $y=22.5$

Thus, the blue flag is on the 5th line at a distance 22.5 m above AB.
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