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**NCERT TEXTBOOK QUESTIONS SOLVED****Page No. 230****(EXERCISE 12.2)**

**Use Ï€ = 22/7 (unless stated otherwise)****Q 1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60Â°.****Sol. **Here,

r = 6 cm

Î¸ = 60Â°

âˆ´ Using, the Area of a sector =

We have,

Area of the sector with r = 6 cm and Î¸ = 60Â°**Q 2. Find the area of a quadrant of a circle whose circumference is 22 cm.****Sol. **Let radius of the circle = r

âˆ´ 2Ï€r = 22

â‡’

Here Î¸ = 90Â°

âˆ´ Area of the quadrant of the circle,**Q 3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.****Sol. **[Length of minute hand] = [radius of the circle]

â‡’ r = 14 cm

âˆµ Angle swept by the minute hand in 60 minutes = 360Â°

âˆ´ Angle swept by the minute hand in 5 minutes =

Now, area of the sector with r = 14 cm and Î¸ = 30Â°

Thus, the required area swept by the minute hand by 5 minutes = 154/3 cm^{2}.**Q 4. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding (i) minor segment (ii) major sector. (Use Ï€ = 3.14)****Sol. **Given, radius of the circle = 10 cm

Angle subtend by chord at centre = 90Â° ...(i)

(i) Area of the minor segment = (Area of the sector OAB) - (Area of Î”AOB formed with radius and chord)

= 3.14 x 25 - 50 = 78.5 - 50 = 28.5 cm^{2}

(ii) Area of the major segment = Area of the circle - Area of the minor segment

= Ï€r^{2} - 28.5 = 3.14 x 10 x 10 - 28.5

= 314 - 28.5 = 285.5 cm^{2}**Q 5. In a circle of radius 21 cm, an arc subtends an angle of 60Â° 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 **

Here, radius = 21 cm and Î¸ = 60Â°

(i) Circumference of the circle = 2Ï€r

(ii) Area of the sector with sector angle 60Â°

(iii) Area of the segment formed by the corresponding chord - area of the sector - area of the Î” formed between chord and radius of the circle

Radius of the circle = 15 cm

Central angle subtends by chord = 60^{0}

Area of sector =

= 117.75 cm^{2}

Area of the triangle formed by radii and chord

Area of the minor segment = Area of the sector -Area of the triangle formed by radii and chord

= 117.75-97.31 =20.44 cm^{2}

Area of the circle = Ï€r^{2}

= 3.14 x 15 x 15 = 706.5 cm^{2}

Area of the major segment = Area of the circle - Area of the minor segment

= 706.5 - 20.44 = 686.06 cm^{2}**Q 7. A chord of a circle of radius 12 cm subtends an angle of 120Â° at the centre. Find the area of the corresponding segment of the circle. (Use Ï€ = 3.14 and âˆš3 = 1.73)****Sol. **Here, Î¸ = 120Â° and r = 12 cm

In Î” OAB, âˆ O = 120Â°

â‡’âˆ A + âˆ B = 180Â° âˆ’ 120 = 60Â°

âˆµ OB = OA = 12 cm â‡’âˆ A = âˆ B = 30Â°

So,

â‡’

In right Î” AMO, 12^{2} âˆ’ 6^{2} = AM^{2}

â‡’ 144 âˆ’ 36 = AM^{2}

â‡’ 108 = AM^{2}

â‡’

â‡’

â‡’

Now, from (2),

= 36 Ã— 1.73 cm^{2} = 62.28 cm^{2} ...(3)

From (1) and (3)

Area of the minor segment = [Area of minor segment] âˆ’ [Area of Î” AOB]

= [150.72 cm^{2}] âˆ’ [62.28 cm^{2}] = 88.44 cm^{2}.**Q 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 figure). Find:(i) the area of that part of the field in which the horse can graze.(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use Ï€ = 3.14)**

âˆ´ Radius of the circular region grazed by the horse = 5 m

(i) Area of the circular portion grazed

[âˆµ Î¸ = 90Â° for a square field.]

(ii) When length of the rope is increased to 10 m,

âˆ´ r = 10 m

â‡’ Area of the circular region where Î¸ = 90Â°.

âˆ´ Increase in the grazing area = 78.5 âˆ’ 19.625 m

âˆ´

(i) Circumference =

Length of 1 piece of wire used to make diameter to divide the circle into 10 equal sectors = 35 mm

âˆ´ Length of 5 pieces = 5 Ã— 35 = 175 mm

âˆ´ Total length of the silver wire = 110 + 175 mm = 285 mm

(ii) Since the circle is divided into 10 equal sectors,

âˆ´ Sector angle

â‡’ Area of each sector

Since circle is divided in 8 equal parts,

âˆ´ Sector angle corresponding to each part

â‡’ Area of a sector (part)

âˆ´ The required area between the two ribs

Sector angle (Î¸) = 115Â°

âˆ´ Area cleaned by each sweep of the blades

[âˆµ Each sweep will have to and fro movement]

**Q 12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80Â° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use Ï€ = 3.14)****Sol. **Here, Radius (r) = 16.5 km

Sector angle (Î¸) = 80Â°

âˆ´ Area of the sea surface over which the ships are warned**Q 13. A round table cover has six equal designs as shown in Fig. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs 0.35 per cm ^{2}. (Use âˆš3 = 1.7)**

Since, the circle is divided into six equal sectors.

âˆ´ Sector angle

âˆ´ Area of the sector with Î¸ = 60Â° and r = 28 cm

Now, area of 1 design

= Area of segment APB

= Area of sector âˆ’ Area of Î”AOB ...(2)

In Î”AOB, âˆ AOB = 60Â°, OA = OB = 28 cm

âˆ´ âˆ OAB = 60Â° and âˆ OBA = 60Â°

â‡’ Î”AOB is an equilateral triangle.

â‡’ AB = AO = BO

â‡’ AB = 28 cm

Draw OM âŠ¥ AB

âˆ´ In right Î”AOM, we have

â‡’

â‡’

...(3)

Now, from (1), (2) and (3), we have:

Area of segment APQ = 410.67 cm^{2} âˆ’ 333.2 cm^{2} = 77.47 cm^{2}

â‡’ Area of 1 design = 77.47 cm^{2}

âˆ´ Area of the 6 equal designs = 6 Ã— (77.47) cm^{2}

= 464.82 cm^{2}

Cost of making the design at the rate of Rs 0.35 per cm^{2},

= Rs 0.35 Ã— 464.82

= Rs 162.68.**Q 14. Tick the correct answer in the following: Area of a sector of angle p (in degrees) of a circle with radius R is****Sol.** Here, radius (r)= R

Angle of sector (Î¸)= pÂ°

âˆ´ Area of the sector

Thus, the option is correct.

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