NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.2 Question 3
NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.2 Question 3
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 (1/4)th the distance AD on the 2nd line and (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?
Given:
- Distance between lines = 1 m
- Number of flower pots along AD = 100
- Distance between flower pots = 1 m
- Total length of AD = 100 m
- Niharika posts a red flag at \( \frac{1}{4} \)th of AD on the 2nd line
- Niharika posts another red flag at \( \frac{1}{5} \)th of AD on the 8th line
To Find:
- The distance between the two red flags
- The position where Rashmi should post a blue flag exactly halfway between the two red flags
Formula:
- Distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ - Midpoint of the segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \):
$$ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
Solution:
Step 1: Let us assign coordinates. Take A as (0, 0). Along AD, the y-coordinate varies from 0 to 100. Each vertical line is parallel to AD and 1 m apart.
Step 2: The 2nd line is at \( x = 2 \). Niharika posts the first red flag at \( \frac{1}{4} \)th of AD, so y-coordinate is \( \frac{1}{4} \times 100 = 25 \).
$$ \text{First flag coordinates: } (2, 25) $$Step 3: The 8th line is at \( x = 8 \). Niharika posts the second red flag at \( \frac{1}{5} \)th of AD, so y-coordinate is \( \frac{1}{5} \times 100 = 20 \).
$$ \text{Second flag coordinates: } (8, 20) $$Step 4: Find the distance between the two flags using the distance formula.
$$ d = \sqrt{(8 - 2)^2 + (20 - 25)^2} = \sqrt{6^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61} $$Step 5: Find the midpoint between the two flags for Rashmi's blue flag.
$$ \left( \frac{2 + 8}{2}, \frac{25 + 20}{2} \right) = (5, 22.5) $$Step 6: Therefore, Rashmi should post the blue flag at the 5th line and 22.5 m from A.
Result:
The distance between the two red flags is \( \sqrt{61} \) m.
Rashmi should post her blue flag at the coordinates (5, 22.5).
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