In order to conduct Sports Day activities in your School, lines

Q) In order to conduct Sports Day activities in your School, lines have been drawn with chalk powder at a distance of 1 m each, in a rectangular shaped ground ABCD, 100 flowerpots have been placed at a distance of 1 m from each other along AD, as shown in given figure below. Niharika runs 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5 th distance AD on the eighth line and posts a red flag.

1. Find the position of green flag
a) (2,25) b) (2,0.25) c) (25,2) d) (0, -25)

2. Find the position of red flag
a) (8,0) b) (20,8) c) (8,20) d) (8,0.2)

3. What is the distance between both the flags?
a). √41 b) √11 c) √61 d) √51

4. If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
a) (5, 22.5) b) (10,22) c) (2,8.5) d) (2.5,20)

5. If Joy has to post a flag at one-fourth distance from green flag ,in the line
segment joining the green and red flags, then where should he post his flag?
a) (3.5,23.75) b) (0.5,12.5) c) (2.25,8.5) d) (25,20)

Ans:

1. Position coordinates of Green flag:

As we can see in the given diagram, Green flag is on 2^nd line, hence its X – coordinate is: 2

On 2^nd line, it has moved $\frac{1}{4}$ ^th of the 100 m distance, therefore, its Y-coordinate is: $\frac{100}{4}$ = 25

∴ the position coordinates of Green Flag = (2, 25)

Hence, option (a) is correct.

2. Find the position of Red flag

As we can see in the given diagram, Red flag is on 8^thline, hence its X – coordinate is: 8

On 8^thline, it has moved $\frac{1}{5}$ ^th of the 100 m distance, therefore, its Y-coordinate is: $\frac{100}{5}$ = 20

∴ the position coordinates of Green Flag = (8, 20)

Hence, option (c) is correct.

3. Distance between the flags:

We have just calculated the coordinates for both of the flags, these are as under:

Green Flag: (2, 25) and Red Flag: (8, 20)

Now the distance between two coordinates is given by:

$\sqrt {(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$

By substituting the values, we get the distance between the two flags as:

$\sqrt {(8 - 2)^2 + (25 - 20)^2}$

= $\sqrt {(36 + 25)} = \sqrt{61}$

Hence, option (c) is correct.

4. Position coordinates of Blue Flag:

The Blue flag is to be posted in halfway.

The coordinates of the two flags are: (2, 25) and (8, 20)

Since, the mid point between the two coordinates are given by:

$(\frac{ X_1 + X_2}{2}, \frac{ Y_1 + Y_2}{2})$

By substituting the values, we get coordinates of the midpoint as:

$(\frac{2 + 8}{2}, \frac{20 + 25}{2})$

= $(\frac{10}{2}, \frac{45}{2})$ = (5, 22.5)

Hence, option (a) is correct.

5. Position coordinates of the 3rd flag, posted at one – fourth distance from green flag:

We just calculated that the coordinates of green flag as (2, 25) and that of Red flag as (8, 20)

Also we calculated the distance between the Green and Red flags is √61.

Since the flag is posted on this line at 1/4 distance from green flag, then the distance of flag from Red Flag is 3/4 of the total line

It clearly means that the new flag divides the line in the ratio of 1:3

Now, If a point divides a line connecting two point (X₁, Y₁) and (X₂, Y₂) in the ratio of m₁: m₂, then the coordinates of this point is given by: $(\frac{m_1 X_2 + m_2 X_1}{m_1 + m_2}, \frac{m_1 X_2 + m_2 X_1}{m_1 + m_2})$