Question

The Olympic symbol, comprising five interlocking rings, represents the union of the five continents of the world and the meeting of athletes from all over the world at the Olympic Games. In order to spread awareness about the Olympic Games, students of Class X took part in various activities organised by the school. One such group of students made 5 circular rings in the school lawn with the help of ropes. Each circular ring required 44 m of rope. Also, in the shaded regions, as shown in the figure, students made rangoli showcasing various sports and games. It is given that ΔOAВ is an equilateral triangle, and all unshaded regions are congruent.

Based on the above information, answer the following questions:

1. Find the radius of each circular ring. (1)
2. What is the measure of ∠AOB? (1)
3. 1. Find the area of the shaded region R₁. (2)
      OR
   2. Find the length of rope around the unshaded regions. (2)

Answer 1

i. Length of rope for 1 ring = 2πr

2πr = 44

`2 xx 22/7 r = 44`

r = `(44 xx 7)/(2 xx 22)`

r = 7m

Radius of each circular ring = 7 m

ii. 

Given that ΔOAB is an equilateral triangle.

∴ ∠AOB = 60°

iii. a. 

Area of shaded region R₁ = Area of circle − Area of unshaded region

= Area of circle − [2 × Area of minor segment]

= πr² − 2 [Area of sector − Area of equilateral triangle]

= `22/7 xx 7 xx 7 - 2 [(πr^2 θ)/360 - (sqrt3)/4 a^2]`

= `154 - 2 [(22/7 xx 7 xx 7 xx 60)/360 - (sqrt3)/4 xx 7 xx 7]`

= `154 - 2 [77/3 - (49sqrt3)/4]`

= `154/1 - 154/3 + (49sqrt3)/2`

= `(462 - 154)/3 + (49sqrt3)/2`

= `308/3 + (49 xx1.732)/2`

= `102.67 + 84.868/2`

= 102.67 + 42.434

= 145.10 m²

OR

b. Length of rope = 8 × length of arc

= `8 xx (2πrθ)/360`

= `(8 xx 2 xx 22/7 xx 7 xx 60)/360`

= `176/3`

= 58.67 m