Areas Related to Circles

NCERT Solution of Exercise 12.2(part2)

Question (7) A chord of a circle of radius 12 cm subtends an angle of 120⁰ at the centre. Find the area of the corresponding segment of the circle (Use π 3.14 and √ 3 = 1.73).

Solution

[Strategy to solve the question (a) Find the area of sector formed by angle 120⁰ (b) Find the area formed by corresponding chord at the centre. (c) Subtract the area of triangle from area of sector to find the area of corresponding segment, i.e. minor segment to find the required answer. ]

Given, Radius of circle = 12 cm

Angle of sector = 120⁰

Area of segment (minor) =?

We know that, area of sector of angle, θ

Therefore, Area of sector of angle 120⁰

= 150.72 cm²

Calculation of area of triangle OAB

Now, in triangle OAB

∠ AOB = 120⁰

OA = OB = 12 cm [Because these are radius of same circle and hence are equal]

Now, a perpendicular OM⊥AB is drawn

Now, in Δ OAM and Δ OMB

OA = OB [radii of same circle]

∠ OMA = ∠ OMB = 90⁰

[Because OM⊥AB]

OM = OM [common side in both of the triangles]

Therefore, according to RHS congruence criterion

Δ OAB ≅ Δ OBM

Therefore, by CPCT (Corresponding pars of congruence triangles are equal)

∠ AOM = ∠ MOB

And, AM = BM

i.e. M is the mid-point of ∠AOB and base, AB

Therefore, ∠ AOM = ∠ MOB = 60⁰

Now, in triangle OAM

∠ OAM = 60⁰

OA = 12 cm

OM = ?

Therefore, now

Now, in Δ OAM,

Since, AM = MB

Therefore, MB = 6 √ 3 cm

Therefore,

AB = AM + MB

⇒ AB= 6 √ 3 cm + 6 √ 3 cm

⇒ AB = 12√3 cm

Now, in triangle OAB

AB = Base = 12 √3

OM = height = 6 cm

We know that, area of a triangle = 1/2 × Base × Height

Thus, Area of triangle OAB = 1/2 × 12√ 3 cm × 6 cm

= 36√ 3 cm²

= 36 × 1.73 cm²

= 62.28 cm²

Now, Area of segment APB

= Area of sector OAPB – Area of triangle OAB

= 150.72 cm² – 62.28 cm²

= 88.44 cm²

Thus, area of segment APB = = 88.44 cm² Answer

Question (8) A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by mans 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 are if the rope were 10 m long instead of 5 m. (Use π = 3.14)

Solution

[Strategy to solve the question (a) Find the area of sector when radius = 5 m (b) Find the area of sector with increase length of rope. (c) Find the difference between two areas to find the required answer.]

Given, Angle of sector = 90⁰

[Because given area is a quadrant (1/4 th part) of the circle]

Area of part of field in which horse can graze, i.e. area of sector when radius = 5m = ?

Increase in grazing area when rope became 10 m long, i.e. increase in area of sector when radius = 10 m = ?

(i) Calculation of Area of grazing when length of rope, i.e. radius of circle = 5 m

We know that area of sector of angle, θ

Therefore, area of sector of angle, 90⁰

= 19.625 cm²

Thus, Area of part of field in which horse can graze = 19.625 cm² Answer

(ii) Increase in grazing area if the rope became 10 m long instead of 5 m

Here, radius = 10 m

Now, Area of sector of angle, 90⁰

= 3.14 × 25 cm²

= 78.5 cm²

Now, increase in grazing area = Grazing area when rope is 10 m long – grazing area when rope was 5 m long

= 78.5 cm² – 19.625 cm²

= 58.875 cm²

Thus, Increase in grazing area if the rope were 20 m long instead of 5 m = 58.875 cm² Answer

Question (9) A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in figure. Find

(i) The total length of the silver wire required

(ii) The area of each sector of the brooch

Solution

[Strategy to solve the question (a) Calculate the circumference of circle (b) Calculate total length of 5 diameters by multiplying the given diameter by 5 (c) Add length of circumference and total length of 5 diameters to find length of silver wire. (d) Find the angle of one of 10 sector by dividing total angle of circle by 10 (e) Find the area of one sector or Find the area of circle and divide it by 10 to find the area of one sector]

Given, Diameter of circle = 35 mm

Therefore, Radius of circle = 35/2 = 17.5 mm

Total number of sector = 10

Total number of diameter = 5

Therefore, length of wire required = ?

Area of each sector = ?

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

We know that circumference of circle = 2 π r

Therefore, circumference of given circle (brooch)

Now, leght of 5 diameter = diameter × 5

= 35 × 5 = 175 mm

Therefore, total length of silver wire

= circumference + total length of diameter

= 110 cm + 175 mm

= 285 mm

Therefore, total length of silver wire required = 285 mm

(ii) Angle subtend by one sector out of 10

Thus, angle subtended by one sector = 36⁰

We know that area of sector of angle, θ

= 96.25 mm²

Therefore, (i) Total length of wire required = 285 mm And (ii) Area of each sector of brooch = 96.25 mm² Answer

Question (10) An umbrella has 8 ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

Solution

[Strategy to solve the question Find the angle subtend by one sector, i.e. by dividing total angle of circle by 8 (b) Find the area of sector]

Given, radius of rib = 45 cm

Total number of ribs = 8

Therefore, angle subtend by one rib = 360/8 = 45⁰

This, means, θ = 45⁰

Area of sector = ?

Now, we know that area of sector of angle θ

= 795.53 cm²

Thus, area between two consecutive ribs of the umbrella = 795.53 cm² Answer

Question (11) A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115⁰. Find the total area cleaned at each sweep of the blades.

Solution

[Strategy to solve the question: Find the area of sector of given angle of 115⁰ (b) Multiply the area by 2 for two blades which will give the total area cleaned at each sweep of the blades.]

Given, length of the blade of wiper = 25 cm

This means radius, r = 25 cm

Angle of sector, θ = 115⁰

Therefore, Area cleaned by two blades in one swipe = ?

We know that area of sector of angle, θ

Therefore, area of sector of angle, 115⁰

Area of sector of angle, 115⁰ = 627.48 cm²

This means area wiped by one blade in one swipe = 627.48 cm²

Therefore, area covered by 2 wiper blades

= Area covered by one blade in one swipe × 2

= 627.48 cm² × 2

= 1254.96 cm²

Thus, total area cleaned at each swipe of the blade = 1254.96 cm² Answer

Question (12) To warn ships for underwater rocks, a lighthouse spreads a red colored light over a sector of angle 80⁰ to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use π = 3.14)

Solution

[Strategy to solve the question (a) Take distance as radius 9b) Find the area of sector of given angle, which will give the area of the sea over which the ships are warned.]

Given, Angle of sector, θ = 80⁰

Distance of spread of colored light, i.e. radius of circle = 16.5 km

Area of sea over which ships are warned, i.e. area of sector = ?

We know that area of sector of angle, θ

Therefore, area of sector of angle of 80⁰

= 189.97 km²

Therefore, Area of the sea over which ships are warned = 189.97 km² Answer

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

Solution

[Strategy to solve the question (a) Find the angle of one sector out of six by dividing angle of circle by 6 (b) Find the area of one sector (c) Find the area of one triangle form by chord (d) Subtract the area of triangle from the area of one sector which will give the area of one section of design (e) Multiply the area of one section of design with 6 to find the area of total design of table (f) Find the cost of design by multiplying the given rate]

Given, Radius of cover, i.e. radius of circle = 28 cm

Total number sector = 6

Therefore, Angle of one sector = Angle of circle/6

= 360/6 = 60⁰

Rate of making of design = Rs 0.35 per cm²

Therefore, total cost of making of design = ?

We know that, Area of sector of angle, θ

Therefore, area of sector of angle, 60⁰

= 410.66 cm²

Calculation of Area of triangle OAB

Since, angle of triangle formed at the centre = 60⁰ and two sides are equal, thus formed angle OAB is an equilateral triangle.

Side of the triangle = 28 cm

Now, we know that area of an equilateral triangle,

Therefore, Area of triangle, OAB

= 333.20 cm²

Now, area of segment, APB

= Area of sector OAPB – Area of triangle OAB

= 410.66 cm² – 333.20 cm²

= 77.46 cm²

Thus, area of one segment, i.e. area of one segment of design = 77.46 cm²

Now, Area of 6 segments

= Total area of design = Area of 1 segment × 6

= 77.46 cm² × 6

= 464.76 cm²

Now,

∵ cost of making of 1 cm² of design = Rs 0.35

Therefore, cost of making of 464.76 cm² of design

= Rs 464.76 × 0.35

Thus, cost of making of design = 162.66 Answer

Alternate method to find the cost of making of design

Area of circle = π r²

Therefore, area of given circle

= 2464 cm²

Now, we know that area of an equilateral triangle,

Therefore, Area of triangle, OAB

= 333.20 cm²s

Therefore, Area of 6 triangle = 333.20 × 6 cm²

= 1999.20 cm²

Therefore, Area of design = Area of circle – Area of 6 triangle

= 2464 – 1999.20

= 464.80 cm²

Now, since, cost of making of 1 cm of design = Rs 0.35

Therefore, cost of making of 464.80 cm² of design

= Rs 464.80 × 0.35

= Rs 162.68

Thus, cost of making of design = Rs 162.88 Answer

Question (14) Tick the correct answer in the following:

Area of a sector of angle p (in degrees) of a circle with radius R is

Explanation

Area of sector of angle θ and Radius, r

Therefore, Area of sector p and radius R

Thus, Option is the correct answer.

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