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Question

In a circle of radius 21 cm, an arc subtends and angle of 60 at the centre. Find:
(i) The 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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Solution

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 60o at O.

i) Length of an arc AXB =60360×2π×r

=16×2×227×21

=22cm

ii) Area of sector AOB =60360×π×r2

=16×227×(21)2

=231cm2

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=41×12=21 cm

OC=21cos30=2132 cm

Area of AOB =12×AB×OC

=12×21×2132=44134 cm2

Area of segment (Area of Shaded region) =(23144134) cm2

492794_465205_ans_11110ca3df0f406a9c7feeddb7fc60da.png

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