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Coordinate Geometry — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsCoordinate Geometry5 Marks
Question
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 Figure. Niharika runs $\frac{1}{4}$th the distance AD on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$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?
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Answer

It can be observed that Niharika posted the green flag at $\frac{1}{4}$th of the distance AD i.e., $\frac{1}{4} \times 100 = 25m$ from the starting point of $2^{nd}$​​​​​​​ line. Therefore, the coordinates of this point G is (2, 25)
Similarly, Preet posted a red flag at $\frac{1}{5}$th of the distance AD i.e., $\frac{1}{5} \times 100 = 20m$ from the starting point of $8^{th}​​​​​​​$​​​​​​​ line. Therefore, the coordinates of this point R are (8, 20)
Now we have the positions of posts by Preet and Niharika
According to distance formula, the distance between points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by
$D=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
Distance between these flags by using distance formula,
$D=\sqrt{\left(8-2_{}\right)^{2}+\left(25_{}-{20}\right)^{2}}$
$=\sqrt{36+25} \mathrm {m}$
$=\sqrt{61} \mathrm{m}$
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)
Now by midpoint formula,
$(X, Y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$
$X=\left(\frac{2+8}{2}\right)=5$
$Y=\left(\frac{25+20}{2}\right)=22.5$
Hence, A (X,Y) = (5, 22.5)
Therefore, Rashmi should post her blue flag at 22.5m on the $5^{th}$ line.

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