# Class 10 RD Sharma- Chapter 15 Areas Related to Circles – Exercise 15.3

### Question 1. AB is a chord of a circle with centre O and radius 4 cm. AB is of length 4 cm and divides the circle into two segments. Find the area of the minor segment.

Solution:

Given,

Radius of the circle with centre O, given by OA and OB = r = 4 cm

Also, length of the given chord AB = 4 cm

Since all the sides are equal, OAB is an equilateral triangle where the angle AOB = 60Â°

Thus, the angle subtended at the centre Î¸ is equal to 60Â°

Now,

Area of the minor segment = (Area of sector) â€“ (Area of triangle AOB)

Therefore, now the required area of the segment is given by (8Ï€/3 â€“ 4âˆš3) cm2

### Question 2. A chord PQ of length 12 cm subtends an angle 120o at the centre of a circle. Find the area of the minor segment cut off by the chord PQ.

Solution:

We have, âˆ POQ = 120o and the line segment PQ = 12 cm

Draw the line segment OV âŠ¥ PQ,

Now,

PV = PQ Ã— (0.5) = 12 Ã— 0.5 = 6 cm

Since, âˆ POV = 120o

Also,

âˆ POV = âˆ QOV = 60o

We have,

In triangle OPQ, we have

Sin Î¸ = PV/ OA

Sin 60o = 6/ OA

Substituting the values, we get,

âˆš3/2 = 6/ OA

OA = 12/ âˆš3 = 4âˆš3 = r

Now, calculating the area of the minor segment

Area of the segment = area of sector OPUQO â€“ area of â–³OPQ

= Î¸/360 x Ï€r2 â€“ Â½ x PQ x OV

= 120/360 x Ï€(4âˆš3)2 â€“ Â½ x 12 x 2âˆš3

= 16Ï€ â€“ 12âˆš3 = 4(4Ï€ â€“ 3âˆš3)

Hence,

Area of the minor segment = 4(4Ï€ â€“ 3âˆš3) cm2

### Question 3. A chord of a circle of radius 14 cm makes a right angle at the centre. Find the areas of the minor and major segments of the circle.

Solution:

Given,

Radius (r) of the circle is given by = 14 cm

Angle subtended by the chord with the centre of the circle, Î¸ is given by 90Â°

Now,

Area of minor segment = Î¸/360 x Ï€r2 â€“ Â½ x r2 sin Î¸

Substituting the values, we get,

= 90/360 x Ï€(14)2 â€“ Â½ x 142 sin (90Â°)

= Â¼ x (22/7) (14)2 â€“ 7 Ã— 14

= 56 cm2

Area of circle = Ï€r2

Substituting the values,

= 22/7 x (14)2 = 616 cm2

Thus, we know,

The area of the major segment = Area of circle â€“ Area of the minor segment

= 616 â€“ 56 = 560 cm2

### Question 4. A chord 10 cm long is drawn in a circle whose radius is 5âˆš2 cm. Find the area of both the segments.

Solution:

Given,

Radius of the circle, r = 5âˆš2 cm = OA = OB

Length of the chord AB = 10 cm

Now,

In triangle OAB,

AB = 100

Thus, Pythagoras theorem is satisfied, that is, OAB is a right-angled triangle.

Angle subtended by the chord at the centre of the circle, Î¸ = 90Â°

We know,

Area of minor segment = area of sector â€“ area of triangle

Substituting the values,

= Î¸/360 Ã— Ï€r2 â€“ Â½ x r2 sin Î¸

= 90/360 Ã— (3.14) 5âˆš22 â€“ Â½ x (5âˆš2)2 sin 90

= [Â¼ x 3.14 x 25 x 2] â€“ [Â½ x 25 x 2 x 1]

We obtain,

= 25(1.57 â€“ 1)

= 14.25 cm2

Area of circle = Ï€r2 = 3.14 x (5âˆš2)2 = 3.14 x 50 = 157 cm2

Thus,

Area of major segment = Area of circle â€“ Area of minor segment

= 157 â€“ 14.25

= 142.75 cm2

### Question 5. A chord AB of circle of radius 14 cm makes an angle of 60Â° at the centre of a circle. Find the area of the minor segment of the circle.

Solution:

Given,

Radius of the circle (r) is given by= 14 cm = OA = OB

Angle subtended by the chord at centre of the circle, Î¸ = 60Â°

In triangle AOB, since, angle opposite to equal sides are equal,

Therefore, angle A = angle B

By angle sum property,

âˆ A + âˆ B + âˆ O = 180

x + x + 60Â° = 180Â°

2x = 120Â°, x = 60Â°

Since, all the angles are 60Â° each, therefore, the triangle OAB is an equilateral with OA = OB = AB

We know,

Area of the minor segment = area of sector â€“ area of triangle OAB

= Î¸/360 Ã— Ï€r2 â€“ 1/2 r2 sin Î¸

= Î¸/360 Ã— Ï€r2 â€“ 1/2 x (14)2 sin 60Â°

= 60/360 Ã— (22/7) 142 â€“ 1/2 x (14)2 x âˆš3/2

= 60/360 x (22/7) 142 â€“ âˆš3/4 x (14)2

= 142 [(1/6) x (22/7)] â€“ 0.4330

= 142 [(22/42)] â€“ 0.4330

= 142 [(11-9.093)/21]

= 142 [0.09080]

= 17.79

Therefore, the area of the minor segment = 17.79 cm2

### Question 6. Find the area of the minor segment of a circle of radius 14 cm, when the angle of the corresponding sector is 60Â°. [Ncert Exemplar]

Solution:

We have,

Radius of the circle, r = 14 cm

Central angle of the corresponding sector = 60Â°

Since, OA = OB = r, therefore, triangle AOB is an isosceles triangle.

âˆ OAB = âˆ OBA =  Î¸

Now, in triangle OAB, by angle sum property

âˆ AOB + âˆ OAB + âˆ OBA = 180Â°

Substituting values,

60Â° +  Î¸ + Î¸ = 180Â°

Solving,

Î¸ = 60Â°

Now, since all the angles are equal , therefore it is an equilateral triangle, with equal sides, that is,

OA = OB = AB = 14 cm

Area of triangle OAB =

Substituting the values, we get

Area = 49 âˆš3 cm2

Also,

Area of sector OBAO =

Now,

Area of minor segment = Area of sector OBAO – Area of triangle OAB =

### Question 7. A chord of a circle of radius 20 cm subtends an angle of 90Â° at the centre. Find the area of the corresponding major segment of the circle. (Use Ï€ = 3.14) [Ncert Exemplar]

Solution:

Let AB be the chord of a circle of radius 10 cm, with O as the centre of the circle

We know,

âˆ AOB = 90Â°

Therefore,

Angle of the major sector = 360Â° – 90Â° = 270Â°

Therefore,

Area of the major sector =

Now,

Draw OM perpendicular to AB

So,

AM = 1/2 AB and âˆ AOM = 1/2Ã—90Â° = 45Â°

Now,

So,

AM = 10/âˆš2

Therefore, AB = 10âˆš2 cm and also, OM=OA

So,

Area of triangle OAB = 1/2 * base * height

Area of the required major segment = 235.5 cm2 + 50 cm2 = 285.5 cm2

Also,

Area of triangle OAB = 1/2 Ã— OA Ã— OB

= 1/2 Ã— 10 Ã— 10 cm2 = 50 cm2

### Question 8. The radius of a circle with centre O is 5 cm (see figure). Two radii OA and OB are drawn at right angles to each other. Find the areas of the segments made by the chord AB (Take Ï€ = 3.14).

Solution:

Radius of the circle ,r = 5 cm

Since, OA and OB are at right angle

Therefore, âˆ AOB = 90Â°

âˆµ  Chord AB makes two segments which are minor segment and major segment respectively.

Now,

Area of the minor segment

Area of circle = pi * r2 = 78.50 cm2

Therefore, area of major segment = 71.375 sq. units.

### Question 9. AB is the diameter of a circle, centre O. C is a point on the circumference such that âˆ COB = 0. The area of the minor segment cut off by AC is equal to twice the area of the sector BOC. Prove that.

Solution:

Given,

âˆ AOC = 180Â° – Î¸

Let us assume r to be the radius of the circle.

Area of sector BOC =

Now,

Area of minor segment AC = 2 * Area of sector BOC

But, Area of segment=

Now,

Also,

### Question 10. A chord of a circle subtends an angle of 0 at the centre of the circle. The area of the minor segment cut off by the chord is one eighth of the area of the circle. Prove that

Solution:

Let chord AB subtends an angle of 0Â° at the centre of a circle with radius r

Now area of the circle = nr1 and area of the minor segment ACB.

But.

Area of the minor segment ACB = 1/8 = area of circle

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