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Q3:
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 $\frac{1}{4}$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?
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 $\frac{1}{4}$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?
Solution :
Given:
1. The school ground is rectangular, represented by a coordinate plane where the x-axis represents the lines (1 to 10) and the y-axis represents the distance along AD (in meters).
2. Total distance along AD = $100$ meters (since there are 100 flower pots placed 1m apart).
3. Niharika runs on the 2nd line ($x_1 = 2$) and covers $\frac{1}{4}$ of the distance AD.
4. Preet runs on the 8th line ($x_2 = 8$) and covers $\frac{1}{5}$ of the distance AD.
To Find:
1. The distance between the green flag (Niharika) and the red flag (Preet).
2. The coordinates where Rashmi should post her blue flag (the midpoint of the two flags).
Step 1: Determine the coordinates of the flags.
Let the position of the green flag be $G(x_1, y_1)$.
$x_1 = 2$
$y_1 = \frac{1}{4} \times 100 = 25$
So, $G = (2, 25)$.
Let the position of the red flag be $R(x_2, y_2)$.
$x_2 = 8$
$y_2 = \frac{1}{5} \times 100 = 20$
So, $R = (8, 20)$.
Step 2: Calculate the distance between the two flags.
Using the Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$d = \sqrt{(8 - 2)^2 + (20 - 25)^2}$
$d = \sqrt{(6)^2 + (-5)^2}$
$d = \sqrt{36 + 25}$
$d = \sqrt{61}$
$d \approx 7.81$ meters.
Step 3: Determine the position of the blue flag.
The blue flag is at the midpoint $M(x, y)$ of the line segment joining $G(2, 25)$ and $R(8, 20)$.
Using the Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
$x = \frac{2 + 8}{2} = \frac{10}{2} = 5$
$y = \frac{25 + 20}{2} = \frac{45}{2} = 22.5$
Therefore, the blue flag should be posted on the 5th line at a distance of 22.5 meters from AD.
Final Answer: The distance between the two flags is $\sqrt{61}$ m (approx 7.81 m). Rashmi should post her blue flag at the coordinates (5, 22.5).
More Questions from Class 10 Mathematics Coordinate geometry EXERCISE 7.2
- Q1: Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.
- Q10: Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order. [Hint : Area of a rhombus = $\frac{1}{2}$ (product of its diagonals)]
- Q2: Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
- Q4: Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
- Q5: Find the ratio in which the line segment joining A(1, – 5) and B(– 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.
- Q6: If (1, 2), (4, $y$), ($x$, 6) and (3, 5) are the vertices of a parallelogram taken in order, find $x$ and $y$.
- Q7: Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4).
- Q8: If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = $\frac{3}{7}$ AB and P lies on the line segment AB.
- Q9: Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
CBSE Solutions for Class 10 Mathematics Coordinate geometry
Chapters in CBSE - Class 10 Mathematics
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