Question

NSS (National Service Scheme) aims to connect the students to the community and to involve them in problem solving process.

NSS symbol is based on the 'Rath' wheel of the Konark Sun Temple situated in Odisha. The wheel signifies the progress cycle of life. The diagrammatic representation of the symbol is given below

Observe the figure given above. The diameters of inner circle are equally placed. Given that OP = 21 cm and OS = 10 cm:

Based on the above information, answer the following questions:.

1. Find m ∠ROS.     [1]
2. Find the perimeter of sector OPQ.     [1]
3. 1. Find the area of shaded region PQRS.     [2]
      OR
   2. Find the area of shaded region ACB, i.e., the segment ACB.     [2]

Answer 1

Given, OP = 21 cm and OS = 10 cm

(i) The inner circle is divided into 8 equal sectors.

Central angle of circle = 360°

∴ m ∠ROS = 360° = 45°

(ii) We know that the perimeter of sector OPQ

= `θ/(360°) xx 2πr + 2r` 

= `(45)/(360) xx 2 xx 22/7 xx 21 + 2 xx 21`  ...[∵ Given OP = 21 cm and m ∠ROS = 45°]

= 16.5 cm + 42 cm

= 58.5 cm

(iii)

(a) Area of shaded region PQRS

= Area of sector OPQ − Area of sector ORS

= `θ/(360°) xx π(OP)^2 - θ/360° xx π(OS)^2`

=  `θ/(360°) xx π[(OP)^2 - (OS)^2]`

= `45/360 xx 22/7 [(21)^2 - (10)^2]`

= `1/8 xx 22/7 [441 - 100]`

= `1/8 xx 22/7 xx 341`

= 133.96 cm²

OR

(b) From figure, we have

∠AOB = 90°

∴ Area of segment ACB

= `π(OA)^2 (90°)/(360°) - "area of"  Δ(AOB)`

= `π xx 100 xx 1/4 - 1/2 xx 10 sin 45° xx  2 xx 10 cos 45°`

= `25π − 100 xx 1/2`

= 25π − 50

= 25(π − 2)

= `25(22/7 − 2)`

= `25 xx 8/7`

= `200/7  "cm"^2`