Question

1. Find the length of an arc of a sector of a circle whose radius is 21

cm and the angle subtended at the centre is 30.
.​

Answer 1

Step-by-step explanation:

In the mentioned figure,

O is the centre of circle,

AB is a chord

AXB is a major arc,

OA=OB= radius =21 cm

Arc AXB subtends an angle 60

o

at O.

i) Length of an arc AXB =

360

60

×2π×r

=

6

1

×2×

7

22

×21

=22cm

ii) Area of sector AOB =

360

60

×π×r

2

=

6

1

×

7

22

×(21)

2

=231cm

2

iii) Area of segment (Area of Shaded region) = Area of sector AOB− Area of △AOB

By trigonometry,

AC=21sin30

OC=21cos30

And, AB=2AC

∴ AB=42sin30=41×

2

1

=21 cm

∴ OC=21cos30=

2

21

3

cm

∴ Area of △ AOB =

2

1

×AB×OC

=

2

1

×21×

2

21

3

=

4

441

3

cm

2

∴ Area of segment (Area of Shaded region) =(231−

4

441

3

) cm

2