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Areas Related to Circles

Short Answer Type

71. The numerical difference between circum-ference and diameter is 30 cm. What is the radius of the circle?

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72. If an arc forms 90° at the centre O of the circle, what is the ratio of its length to the circumference of the circle?

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73. Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major segment. (use π = 3.14)

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74. Find the area of the segment of a circle of radius 14 cm, if the length of the corresponding arc APB is 22 cm. [Use π = 22/7]

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75. AB is a chord of a circle of radius 10 cm. The chord subtends a right angle at the centre of the circle. Find the area of the minor segment.

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76.
In a circle of radius 21 cm, an arc subtends an angle 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.

Here, we have
r = 21 cm
and ө = 60°

Here, we haver = 21 cmand ө = 60°Area of circle = Area of se

Area of circle = $πr squared$

$space equals space open parentheses 22 over 7 straight x space 21 space straight x space 21 close parentheses space cm squared space equals space open parentheses 22 space straight x space 3 space straight x space 21 close parentheses space cm squared space equals space 1386 space cm squared$

Area of sector OACBO

$equals space πr squared space straight theta over 360 space equals space open parentheses 22 over 7 straight x space 21 space straight x space 21 space straight x space 60 over 360 close parentheses space cm squared equals space left parenthesis 11 space straight x space 21 right parenthesis space cm squared equals space 231 space cm squared space space space space space space$

And, Area of triangle (AOB)

$equals space 1 half space straight r squared space sin space straight theta equals space space 1 half space straight r squared space sin space 60 degree equals space space open parentheses 1 half space straight x space 21 space straight x space 21 space straight x space fraction numerator square root of 3 over denominator 2 end fraction close parentheses equals space open parentheses fraction numerator 21 space straight x space 21 space straight x space square root of 3 over denominator 4 end fraction close parentheses space equals space open parentheses fraction numerator 441 space straight x space 1.732 over denominator 4 end fraction close parentheses equals space fraction numerator 763.81 over denominator 4 end fraction space cm squared equals space 190.95 space cm squared$
Now,
(i) The length of the arc $ACB with overbrace on top$

$equals space 2 πr space straight x space straight theta over 360 space equals space open parentheses 2 space straight x space 22 over 7 straight x space 21 space straight x space 60 over 360 close parentheses space cm equals space 22 space cm$

(ii) Area of sector (OACBO) = 231 cm²(iii) Area of minor segment (ACBA)
= Area of circle - Area of (∆AOB)
= (1386 - 190.95) cm²= 1195.05 cm²

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77. A sector is cut off from a circle of radius 21 cm. The angle subtended at centre is 120°, find the length of its arc and the area.

1045 Views

78. A pendulum swings through an angle 30° and describes an arc 8.8 cm in length. Find the length of the pendulum

open parentheses Use space equals space 22 over 7 close parentheses.

1356 Views

79. The minute hand of a clock is 12 cm long. Find the area of the face of the clock described by minute hand in 35 minutes.

297 Views

80.

Two circles touch externally. The sum of their areas is 130π sq. cm and the distance between their centres is 14 cm. Find the radii of the circle.

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Previous Year Papers

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Short Answer Type

76. In a circle of radius 21 cm, an arc subtends an angle 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.

76.
In a circle of radius 21 cm, an arc subtends an angle 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.