In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. 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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In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. 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 ?

Solution:

⇨ Given

r = 21 cm

θ = 60°

⇨ (i) Length of Arc

$⇨ (i) Length of Arc$

$=\frac{60}{360}\times 2\times \frac{22}{7}\times 21$

$=\frac{1\times 2\times 22 \times 21}{6\times 7}$

$=\frac{22 \times 3}{3}$

$=22 cm$

⇨ (ii) Area of sector formed by Arc

$⇨ (ii) Area of sector formed by Arc$

$=\frac{60}{360}\times \frac{22}{7}\times \left ( 21 \right )^{2}$

$=\frac{22\times 21\times 21}{6\times 7}$

$=\frac{22\times 21\times 3}{6}$

$=\frac{22\times 21}{2}$

$=11\times 21$

$=231 cm^{2}$

⇨Area Of Equilateral △AOB

$⇨Area Of Equilateral △AOB$

$=\frac{\sqrt{3}}{4}\times \left ( 21 \right )^{2}$

$=\frac{441\sqrt{3}}{4}cm^{2}$

⇨ (iii) Area of the segment formed by the corresponding chord

= Area of sector formed by Arc - Area Of Equilateral △AOB

$In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord ?$

Exercise 12.2

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6. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.73)

7. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and √3 = 1.73)

8. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig. 12.11). Find

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(i) the total length of the silver wire required.

(ii) the area of each sector of the brooch

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11. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades.

12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use π = 3.14)

13. A round table cover has six equal designs as shown in Fig. 12.14. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ` 0.35 per cm2 . (Use √3 = 1.7)

14. Tick the correct answer in the following :Area of a sector of angle p (in degrees) of a circle with radius R is

In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. 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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