Question

Read the following passage:

People of a circular village Dharamkot want to construct a road nearest to it. The road cannot pass through the village. But the people want the road at a shortest distance from the centre of the village. Suppose the road starts from A which is outside the circular village (as shown in the figure) and touch the boundary of the circular village at B such that AB = 20 m. Also the distance of the point A from the centre O of the village is 25 m.

Based on the above information, answer the following questions:

1. If B is the mid-point of AC, then find the distance AC.
2. Find the shortest distance of the road from the centre of the village.
3. Find the circumference of the village.
   OR
   Find the area of the village.

Answer 1

i. B is the mid-point of AC

∴ AC = 2AB

AC = 2 × 20 = 40 m

ii. Shortest distance of the road from the centre of circle = Radius of circle

In ΔOAB, ∠B = 90°

∴ OB² + AB² = OA²

OB² + 20² = 25²

OB² = 625 – 400

OB = `sqrt(225)` = 15

∴ Shortest distance = 15 m

iii. Circumference of the village

= 2πr

= `2 xx 22/7 xx 15`

= `660/7`

= `94 2/7 "m"`

OR

iii. Area of the village 

= πr²

= `22/7 xx 15 xx 15`

= `4950/7`

= `707 1/7 "m"^2`