Q) An arc of a circle of radius 21 cm subtends an angle of 60° at the centre. Find:
(i) the length of the arc
(ii) the area of the minor segment of the circle made by the corresponding chord radius r of in-circle.

Ans: Let’s draw the diagram for better understanding:

(i) length of the arc:

We know that the length of the arc is given by =

Here, we are given that θ  = 600, r = 21 cm

Therefore, length of the arc =

= = 22 cm

Therefore the length of the arc is 22 cm

(ii) Area of minor segment:

From the above diagram, Area of minor segment APB

= Area of sector AOBP – Area of triangle AOB

We know that Area of minor segment APB =

Here, we are given that θ  = 600, r = 21 cm

∴ Area of minor segment APB =

= = 11 x 21 = 231 cm2

Next Area of Δ AOB:

Here, we can see that Δ  AOB is a equilateral triangle.

(∠ AOB is 600   Therefore, sum of other two angle sis 1200 . Since sides OA and OB are equal, hence angles opposite to equal sides will also be equal. It makes all angles 600)

Sine area of a equilateral triangle is =

Here, a = 21 cm, Therefore, area of Δ AOB = cm2

Area of minor segment APB = Area of sector AOBP – Area of triangle AOB

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