To conduct Sports Day activities, in your rectangular shaped school ground , lines have been drawn with chalk powder at a distance of each. flower pots have been placed at a distance of from each other along , as shown in Fig. 7.12. Niharika runs the distance on the 2nd line and posts a green flag. Preet runs the distance 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?

Understand the Question Model the rectangular ground as a Cartesian coordinate system with origin at , where lies along the -axis and along the -axis. Along , flower pots are placed apart, making the total distance . The line number gives the -coordinate, and the fraction of distance run along gives the -coordinate. To find the distance between Niharika's and Preet's flags, use the Distance Formula: To find where Rashmi should post her flag halfway between them, use the Midpoint Formula: Step 1 · Find Coordinates of Niharika's Flag Niharika runs on the line, so . Distance along Therefore, Niharika's green flag is at . Step 2 · Find Coordinates of Preet's Flag Preet runs on the line, so . Therefore, Preet's red flag is at . Step 3 · Calculate Distance Between Both Flags Using the distance formula for and : Step 4 · Find Position of Rashmi's Flag Rashmi posts her blue flag at the midpoint of segment : Rashmi's flag is at . Answer > Distance between the flags ; Rashmi should post her flag on the line at a distance of . Common Mistakes Swapping and Coordinates: Confusing the vertical line number with the -coordinate instead of the -coordinate (e.g., writing instead of ). Squaring Negative Numbers: Computing as rather than , leading to instead of . Misinterpreting the Midpoint Location: Forgetting to interpret the final coordinates in physical terms as "on the line at distance".

Coordinate Geometry | Exercise 7.2

Question 3

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

Solution

Understand the Question

Model the rectangular ground $ABCD$ as a Cartesian coordinate system with origin at $A$ , where $AB$ lies along the $x$ -axis and $AD$ along the $y$ -axis.
Along $AD$ , $100$ flower pots are placed $1\text{ m}$ apart, making the total distance $AD = 100\text{ m}$ .
The line number gives the $x$ -coordinate, and the fraction of distance run along $AD$ gives the $y$ -coordinate.
To find the distance between Niharika's and Preet's flags, use the Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
To find where Rashmi should post her flag halfway between them, use the Midpoint Formula: $M = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$

Step 1 · Find Coordinates of Niharika's Flag

Niharika runs on the $2^{\text{nd}}$ line, so $x = 2$ . Question diagram

Distance along $AD = 100\text{ m}$

\begin{aligned} y_{\text{Niharika}} &= \dfrac{1}{4} \times 100 \\[0.6em] &= 25 \end{aligned}

Therefore, Niharika's green flag is at $N(2, 25)$ .

Step 2 · Find Coordinates of Preet's Flag

Preet runs on the $8^{\text{th}}$ line, so $x = 8$ .

\begin{aligned} y_{\text{Preet}} &= \dfrac{1}{5} \times 100 \\[0.6em] &= 20 \end{aligned}

Therefore, Preet's red flag is at $P(8, 20)$ .

Step 3 · Calculate Distance Between Both Flags

Using the distance formula for $N(2, 25)$ and $P(8, 20)$ :

\begin{aligned} \text{Distance } NP &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\[0.6em] &= \sqrt{(8 - 2)^2 + (20 - 25)^2} \\[0.6em] &= \sqrt{(6)^2 + (-5)^2} \\[0.6em] &= \sqrt{36 + 25} \\[0.6em] &= \sqrt{61}\text{ m} \end{aligned}

Step 4 · Find Position of Rashmi's Flag

Rashmi posts her blue flag at the midpoint $M(x, y)$ of segment $NP$ :

\begin{aligned} x &= \dfrac{x_1 + x_2}{2} \\[0.6em] &= \dfrac{2 + 8}{2} \\[0.6em] &= \dfrac{10}{2} \\[0.6em] &= 5 \end{aligned}

\begin{aligned} y &= \dfrac{y_1 + y_2}{2} \\[0.6em] &= \dfrac{25 + 20}{2} \\[0.6em] &= \dfrac{45}{2} \\[0.6em] &= 22.5 \end{aligned}

Rashmi's flag is at $(5, 22.5)$ .

Answer

Distance between the flags $= \sqrt{61}\text{ m}$ ; Rashmi should post her flag on the $5^{\text{th}}$ line at a distance of $22.5\text{ m}$ .

Common Mistakes

Swapping $x$ and $y$ Coordinates: Confusing the vertical line number with the $y$ -coordinate instead of the $x$ -coordinate (e.g., writing $(25, 2)$ instead of $(2, 25)$ ).
Squaring Negative Numbers: Computing $(-5)^2$ as $-25$ rather than $+25$ , leading to $\sqrt{36 - 25} = \sqrt{11}$ instead of $\sqrt{36 + 25} = \sqrt{61}$ .
Misinterpreting the Midpoint Location: Forgetting to interpret the final coordinates $(5, 22.5)$ in physical terms as "on the $5^{\text{th}}$ line at $22.5\text{ m}$ distance".

Question 3

More questions in Exercise 7.2