In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:
(i) length of the arc.
(ii) area of the sector formed by the arc.
(iii) area of the segment formed by the corresponding chord.
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Answers
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
Step-by-step explanation:
Radius (r) of circle = 21 cm
Radius (r) of circle = 21 cmAngle subtended by the given arc = 60°
Radius (r) of circle = 21 cmAngle subtended by the given arc = 60°Length of an arc of a sector of angle θ =
Radius (r) of circle = 21 cmAngle subtended by the given arc = 60°Length of an arc of a sector of angle θ = Length of arc ACB =
Radius (r) of circle = 21 cmAngle subtended by the given arc = 60°Length of an arc of a sector of angle θ = Length of arc ACB == 22 cm
Radius of the circle, r = 21 cm.
Angle at the centre of the circle, θ = 60°
(1) the length of the arc ?
(2) Area of the sector formed by the arc?
(3) Area of the segment.
