Question

To conduct sports day activities in a rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 10 flower pots have been placed at a distance of 10 m from each other along AD, as shown in the figure. Three students run on the lines 3, 5 and 9 respectively. At any instant, they are at the position P, Q and R respectively.

(a) The co-ordinate of the mid-point of the line segment joining P and Q : 

i. (3, 55) 

ii. (4, 55) 

iii. (55, 4)

iv. (4, 60)

(b) The distance \(\frac{1}{\sqrt{26}}\) PR is equal to : 

i. 3 

ii. 4 

iii. 5 

iv. 6

(c) The co-ordinate of centroid of a Δ formed by P, Q and R is : 

i. (\(\frac{20}{3}\),\(\frac{170}{3}\)) 

ii. (\(\frac{17}{3}\),\(\frac{170}{3}\)) 

iii. (\(\frac{170}{3}\),\(\frac{17}{3}\)) 

iv. (\(\frac{170}{3}\), \(\frac{20}{3}\))

(d) The co-ordinate of the point which divides the joining of P & R in the ratio 1 : 2 is : 

i. (4,40) 

ii. (5,40) 

iii. (6,50) 

iv. (5,50)

(e) The co-ordinate of the mid-point of line segment joining P and R is :

i. (6,70) 

ii. (7,70) 

iii. (8,60) 

iv. (7,50)

Answer 1

The co-ordinates of points P, Q and R are P(3, 30), Q(5, 80) and R(9, 60), respectively.

(a) Option : (ii) 

The midpoint of the line segment joining P and Q is 

(\(\frac{3+5}{2}\), \(\frac{30+80}{2}\)) = (4, 55).

(By mid-point formula between two points) 

Hence,

Option (ii) is correct.

(b) Option : (iv)

The distance between points P and R is 

PR =  \(\sqrt{(9-3)^2+(60-30)^2}\) 

(By distance formula between two points)

Therefore,

\(\frac{1}{\sqrt{26}}\)PR

= \(\frac{6\sqrt{26}}{\sqrt{26}}\) 

= 6m.

Hence,

Option (iv) is correct.

(c) Option : (ii) 

The co-ordinate of centroid of triangle ΔPQR is given by,

Hence,

Option (ii) is correct.

(d) Option : (ii) 

Let the point A(x, y) divides line segment joining points P & R in the ratio 1:2.

Hence,

Option (ii) is correct.

(e) Option : (ii) 

The mid-point of line segment joining Q & R is :

Hence, 

Option (ii) is correct.