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Coordinate Geometry
EXERCISE 7.2
CBSE_NCERT_TEXTBOOK
Grade 10
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 Fig. 7.12. Niharika runs 1 4 th the distance AD on the 2nd line and posts a green flag. Preet runs 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?
Step-by-Step Solution
Key Concept: Use a coordinate system to represent the rectangular ground. Apply the distance formula \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) to find the distance between the two flags and the midpoint formula \(M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\) to locate the position of the blue flag.
1. Set up the coordinate system – Take \(A\) as the origin \((0,0)\) and let the side \(AD\) lie on the \(y\)-axis. Since 100 flower pots are placed 1 m apart on \(AD\), the length \(AD = 100\) m, so \(D\) is at \((0,100)\).
2. Identify the vertical lines – The lines drawn at 1 m intervals are parallel to \(AD\). The 2nd line is therefore the line \(x = 1\) and the 8th line is the line \(x = 7\).
3. Coordinates of the green flag (G) – Niharika runs \(\frac{1}{4}\) of \(AD\) i.e. \(\frac{1}{4}\times100 = 25\) m on the 2nd line. Hence \(G\) has coordinates \((1, 25)\).
4. Coordinates of the red flag (R) – Preet runs \(\frac{1}{5}\) of \(AD\) i.e. \(\frac{1}{5}\times100 = 20\) m on the 8th line. Hence \(R\) has coordinates \((7, 20)\).
5. Distance between the two flags – Using the distance formula:
$$\begin{aligned}
GR &= \sqrt{(7-1)^2 + (20-25)^2} \\
&= \sqrt{6^2 + (-5)^2} \\
&= \sqrt{36+25} \\
&= \sqrt{61}\ \text{m} \approx 7.81\ \text{m}
\end{aligned}$$
6. Mid‑point of \(GR\) – The point exactly halfway between the two flags is the midpoint:
$$M\left(\frac{1+7}{2},\frac{25+20}{2}\right) = (4, 22.5)$$
Thus Rashmi should place the blue flag at the point \((4, 22.5)\), i.e. on the 5th vertical line, 22.5 m above point \(A\).
2. Identify the vertical lines – The lines drawn at 1 m intervals are parallel to \(AD\). The 2nd line is therefore the line \(x = 1\) and the 8th line is the line \(x = 7\).
3. Coordinates of the green flag (G) – Niharika runs \(\frac{1}{4}\) of \(AD\) i.e. \(\frac{1}{4}\times100 = 25\) m on the 2nd line. Hence \(G\) has coordinates \((1, 25)\).
4. Coordinates of the red flag (R) – Preet runs \(\frac{1}{5}\) of \(AD\) i.e. \(\frac{1}{5}\times100 = 20\) m on the 8th line. Hence \(R\) has coordinates \((7, 20)\).
5. Distance between the two flags – Using the distance formula:
$$\begin{aligned}
GR &= \sqrt{(7-1)^2 + (20-25)^2} \\
&= \sqrt{6^2 + (-5)^2} \\
&= \sqrt{36+25} \\
&= \sqrt{61}\ \text{m} \approx 7.81\ \text{m}
\end{aligned}$$
6. Mid‑point of \(GR\) – The point exactly halfway between the two flags is the midpoint:
$$M\left(\frac{1+7}{2},\frac{25+20}{2}\right) = (4, 22.5)$$
Thus Rashmi should place the blue flag at the point \((4, 22.5)\), i.e. on the 5th vertical line, 22.5 m above point \(A\).
Correct Answer: Distance between the flags = \(\sqrt{61}\) m (≈ 7.81 m). The blue flag should be placed at \((4,\;22.5)\) – on the 5th line, 22.5 m from \(A\).
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