#### Chapters

Chapter 2: Exponents of Real Numbers

Chapter 3: Rationalisation

Chapter 4: Algebraic Identities

Chapter 5: Factorisation of Algebraic Expressions

Chapter 6: Factorisation of Polynomials

Chapter 7: Linear Equations in Two Variables

Chapter 8: Co-ordinate Geometry

Chapter 9: Introduction to Euclid’s Geometry

Chapter 10: Lines and Angles

Chapter 11: Triangle and its Angles

Chapter 12: Congruent Triangles

Chapter 13: Quadrilaterals

Chapter 14: Areas of Parallelograms and Triangles

Chapter 15: Circles

Chapter 16: Constructions

Chapter 17: Heron’s Formula

Chapter 18: Surface Areas and Volume of a Cuboid and Cube

Chapter 19: Surface Areas and Volume of a Circular Cylinder

Chapter 20: Surface Areas and Volume of A Right Circular Cone

Chapter 21: Surface Areas and Volume of a Sphere

Chapter 22: Tabular Representation of Statistical Data

Chapter 23: Graphical Representation of Statistical Data

Chapter 24: Measures of Central Tendency

Chapter 25: Probability

## Chapter 15: Circles

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 CirclesExercise 15.1 [Pages 5 - 6]

Fill in the blank:

All points lying inside/outside a circle are called .................. points/ .....................points.

Fill in the blank

Circles having the same centre and different radii are called ...........................circles.

Fill in the blank:

A point whose distance from the centre of a circle is greater than its radius lies in ..................... of the circle.

Fill in the blank

A continuous piece of a circle is ............... of the circle

Fill in the blank:

The longest chord of a circle is a ................of the circle.

Fill in the blank:

An arc is a ................ when its ends are the ends of a diameter.

Fill in the blank:

Segment of a circle is the region between an arc and .................. of the circle.

Fill in the blank:

A circle divides the plane, on which it lies, in ............ parts.

true or false

A circle is a plane figure.

true or false

Line segment joining the centre to any point on the circle is a radius of the circle,

true or false

If a circle is divided into three equal arcs each is a major arc.

true or false

A circle has only finite number of equal chords.

true or false

A chord of a circle, which is twice as long is its radius is a diameter of the circle.

true or false

Sector is the region between the chord and its corresponding arc.

true or false

The degree measure of an arc is the complement of the central angle containing the arc.

ture or false v

The degree measure of a semi-circle is 180°.

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 CirclesExercise 15.2 [Pages 28 - 29]

The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.

Find the length of a chord which is at a distance of 5 cm from the centre of a circle ofradius 10 cm.

Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.

Give a method to find the centre of a given circle.

A line segment AB is of length 5 cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your answer.

Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

An equilateral triangle of side 9cm is inscribed in a circle. Find the radius of the circle.

Given an arc of a circle, show how to complete the circle.

Given an arc of a circle, complete the circle.

Suppose You Are Given a Circle. Give a Construction to Find Its Centre.

The lengths of two parallel chords of a circle are 6 cm and 8 cm. if the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?

Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel. If the distance between AB and CD is 3 cm, find the radius of the circle.

Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Prove that two different circles cannot intersect each other at more than two points.

Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are opposite side of its center. If the distance between AB and CD is 6 cm. Find the radius of the circle.

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 CirclesExercise 15.3 [Page 47]

Three girls Ishita, Isha and Nisha are playing a game by standing on a circle of radius 20 m drawn in a park. Ishita throws a ball o Isha, Isha to Nisha and Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is 24 m each, what is the distance between Ishita and Nisha.

A circular park of radius 40 m is situated in a colony. Three boys Ankur, Amit and Anand are sitting at equal distance on its boundary each having a toy telephone in his hands to talk

to each other.

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 CirclesExercise 15.4 [Pages 72 - 74]

In the below fig. O is the centre of the circle. If ∠APB = 50°, find ∠AOB and ∠OAB.

In the below fig. O is the centre of the circle. Find ∠BAC.

If O is the centre of the circle, find the value of x in the following figure:

If O is the centre of the circle, find the value of x in the following figure:

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If O is the centre of the circle, find the value of x in the following figure

If *O* is the centre of the circle, find the value of *x* in the following figures.

If *O* is the centre of the circle, find the value of *x* in the following figures.

*O* is the circumcentre of the triangle *ABC* and *OD* is perpendicular on *BC.* Prove that ∠*BOD* = ∠*A*

In the given figure, O is the centre of the circle, BO is the bisector of ∠ABC. Show that AB = BC.

In the given figure,* O* and *O*' are centres of two circles intersecting at *B* and *C*. *ACD* is a straight line, find *x*.

In the given figure, if ∠*ACB* = 40°, ∠*DPB* = 120°, find ∠*CBD*.

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

In the given figure, it is given that *O* is the centre of the circle and ∠*AOC* = 150°. Find ∠*ABC*.

In the given figure, *O *is the centre of the circle, prove that ∠*x* = ∠*y* + ∠*z. *

In the given figure, *O* is the centre of a circle and *PQ* is a diameter. If ∠*ROS* = 40°, find ∠*RTS*.

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 CirclesExercise 15.5 [Pages 100 - 104]

In the given figure, Δ*ABC* is an equilateral triangle. Find *m*∠*BEC*.

In the given figure, Δ*PQR* is an isosceles triangle with *PQ* = *PR* and *m* ∠*PQR* = 35°. Find *m* ∠*QSR* and *m *∠*QTR*.

In the given figure, *O* is the centre of the circle. If ∠*BOD* = 160°, find the values of* x* and *y*.

In the given figure, *ABCD* is a cyclic quadrilateral. If ∠*BCD* = 100° and ∠*ABD* = 70°, find ∠*ADB*.

If *ABCD* is a cyclic quadrilateral in which *AD* || *BC* (In the given figure). Prove that ∠*B* = ∠*C*.

In the given figure, *O *is the centre of the circle. Find ∠*CBD**.*

In the given figure, *AB* and *CD* are diameters of a circle with centre *O*. If ∠*OBD* = 50°, find ∠*AOC*.

On a semi-circle with *AB* as diameter, a point *C* is taken, so that m (∠*CAB*) = 30°. Find *m*(∠*ACB*) and m (∠*ABC*).

In a cyclic quadrilateral *ABCD* if *AB* || *CD* and ∠*B** *= 70°, find the remaining angles.

In a cyclic quadrilateral *ABCD*, if *m *∠*A* = 3 (*m* ∠*C*). Find *m* ∠*A*.

In the given figure, *O* is the centre of the circle and ∠*DAB* = 50° . Calculate the values of *x*and *y*.

In the given figure, if ∠*BAC** *= 60° and ∠*BCA* = 20°, find ∠*ADC*.

In the given figure, if *ABC* is an equilateral triangle. Find ∠*BDC* and ∠*BEC*.

In the given figure, *O* is the centre of the circle. If ∠*CEA* = 30°, Find the values of *x*, *y* and *z*.

In the given figure, ∠*BAD* = 78°, ∠*DCF* = *x*° and ∠*DEF* =* y*°. Find the values of *x* and *y*.

In a cyclic quadrilateral ABCD, if ∠*A* − ∠*C* = 60°, prove that the smaller of two is 60°

In the given figure, *ABCD* is a cyclic quadrilateral. Find the value of *x*.

*ABCD *is a cyclic quadrilateral in *BC* || *AD*, ∠*ADC* = 110° and ∠*BAC* = 50°. Find ∠*DAC*.

*ABCD *is a cyclic quadrilateral in ∠*DBC* = 80° and ∠*BAC* = 40°. Find ∠*BCD*.

*ABCD *is a cyclic quadrilateral in ∠*BCD* = 100° and ∠*ABD* = 70° find ∠*ADB*.

Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals.

If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are equal.

Circles are described on the sides of a triangle as diameters. Prove that the circles on any two sides intersect each other on the third side (or third side produced).

*ABCD* is a cyclic trapezium with *AD* || *BC*. If ∠*B* = 70°, determine other three angles of the trapezium.

In the given figure, *ABCD* is a cyclic quadrilateral in which *AC* and *BD* are its diagonals. If ∠*DBC* = 55° and ∠*BAC* = 45°, find ∠*BCD**.*

Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

Prove that the centre of the circle circumscribing the cyclic rectangle *ABCD* is the point of intersection of its diagonals.

*ABCD* is a cyclic quadrilateral in which *BA* and *CD* when produced meet in *E* and *EA* = *ED*. Prove that *AD* || *BC . *

*ABCD* is a cyclic quadrilateral in which *BA* and *CD* when produced meet in *E* and *EA* = *ED*. Prove that *EB* = *EC*.

Prove that the angle in a segment shorter than a semicircle is greater than a right angle.

Prove that the angle in a segment greater than a semi-circle is less than a right angle.

Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to its opposite vertex is half the hypotenuse.

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 Circles [Pages 107 - 109]

In the given figure, two circles intersect at *A* and *B*. The centre of the smaller circle is* **O*and it lies on the circumference of the larger circle. If ∠*APB* = 70°, find ∠*ACB*.

In the given figure, two congruent circles with centres *O* and *O*' intersect at *A* and *B*. If ∠*AOB* = 50°, then find ∠*APB*.

In the given figure, *ABCD* is a cyclic quadrilateral in which ∠*BAD* = 75°, ∠*ABD* = 58° and ∠*ADC *= 77°, *AC* and *BD* intersect at *P*. Then, find ∠*DPC*.

In the given figure, if ∠*AOB* = 80° and ∠*ABC* = 30°, then find ∠*CAO. *

In the given figure, *A* is the centre of the circle. *ABCD* is a parallelogram and *CDE *is a straight line. Find ∠*BCD* : ∠*ABE*.

In the given figure, *AB* is a diameter of the circle such that ∠*A* = 35° and ∠*Q* = 25°, find ∠*PBR*.

In the given figure, *P* and *Q* are centres of two circles intersecting at *B* and *C*. *ACD* is a straight line. Then, ∠*BQD* =

In the given figure, if *O* is the circumcentre of ∠*ABC*, then find the value of ∠*OBC* + ∠*BAC*.

If the given figure, AOC is a diameter of the circle and arc AXB = \[\frac{1}{2}\] arc BYC. Find ∠BOC.

In the given figure, *ABCD* is a quadrilateral inscribed in a circle with centre *O*. *CD* is produced to *E* such that ∠*AED* = 95° and ∠*OBA* = 30°. Find ∠*OAC*.

### RD Sharma solutions for Mathematics for Class 9 Chapter 15 Circles [Pages 109 - 112]

If the length of a chord of a circle is 16 cm and is at a distance of 15 cm from the centre of the circle, then the radius of the circle is

15 cm

16 cm

17 cm

34 cm

The radius of a circle is 6 cm. The perpendicular distance from the centre of the circle to the chord which is 8 cm in length, is

- \[\sqrt{5} \] cm
- \[2\sqrt{5}\] cm
\[2\sqrt{7} \] cm

- \[\sqrt{7}\] cm

If O is the centre of a circle of radius r and AB is a chord of the circle at a distance r/2 from O, then ∠BAO =

60°

45°

30°

15°

*ABCD* is a cyclic quadrilateral such that ∠*ADB* = 30° and ∠*DCA* = 80°, then ∠*DAB* =

70°

100°

125°

150°

A chord of length 14 cm is at a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is

12 cm

12 cm

12 cm

18 cm

One chord of a circle is known to be 10 cm. The radius of this circle must be

5 cm

greater than 5 cm

greater than or equal to 5 cm

less than 5 cm

*ABC* is a triangle with *B* as right angle, *AC *= 5 cm and *AB* = 4 cm. A circle is drawn with* A*as centre and *AC *as radius. The length of the chord of this circle passing through *C* and *B* is

3 cm

4 cm

5 cm

6 cm

If *AB*, *BC* and *CD* are equal chords of a circle with *O* as centre and* AD* diameter, than ∠*AOB* =

60°

90°

120°

none of these

Let *C* be the mid-point of an arc *AB* of a circle such that m \[ \stackrel\frown{AB}\] = 183°. If the region bounded by the arc *ACB* and the line segment *AB* is denoted by *S*, then the centre *O* of the circle lies

in the interior of

*S*in the exertior of

*S*on the segment

*AB*on AB and bisects

*AB*

In a circle, the major arc is 3 times the minor arc. The corresponding central angles and the degree measures of two arcs are

90° and 270°

90° and 90°

270° and 90°

60° and 210°

If A and B are two points on a circle such that m \[ \stackrel\frown{AB}\] = 260°. A possible value for the angle subtended by arc BA at a point on the circle is

100°

75°

50°

25°

An equilateral triangle *ABC *is inscribed in a circle with centre *O*. The measures of ∠*BOC*is

30°

60°

90°

120°

If two diameters of a circle intersect each other at right angles, then quadrilateral formed by joining their end points is a

rhombus

rectangle

parallelogram

square

If *ABC* is an arc of a circle and ∠*ABC** *= 135°, then the ratio of arc \[\stackrel\frown{ABC}\] to the circumference is

1 : 4

3 : 4

3 : 8

1 : 2

The chord of a circle is equal to its radius. The angle subtended by this chord at the minor arc of the circle is

60°

75°

120°

150°

*PQRS *is a cyclic quadrilateral such that *PR* is a diameter of the circle. If ∠*QPR* = 67° and ∠*SPR* = 72°, then ∠*QRS* =

41°

23°

67°

18°

If *A* , *B*, *C* are three points on a circle with centre *O *such that ∠*AOB* = 90° and ∠*BOC** *= 120°, then ∠*ABC* =

60°

75°

90°

135°

The greatest chord of a circle is called its

radius

secant

diameter

none of these

Angle formed in minor segment of a circle is

acute

obtuse

right angle

none of these

Number of circles that can be drawn through three non-collinear points is

1

0

2

3

In the given figure, if chords *AB *and *CD* of the circle intersect each other at right angles, then *x* + *y* =

45°

60°

75°

90°

In the given figure, if ∠*ABC* = 45°, then ∠*AOC *=

45°

60°

75°

90°

In the given figure, chords *AD* and *BC* intersect each other at right angles at a point P. If ∠*DAB* = 35°, then

35°

45°

55°

65°

In the given figure, *O* is the centre of the circle and ∠*BDC* = 42°. The measure of ∠*ACB* is

42°

48°

58°

52°

In a circle with centre O, *AB* and* CD* are two diameters perpendicular to each other. The length of chord* AC* is

2

*AB*- \[\sqrt{2}\]
- \[\frac{1}{2}AB\]
- \[\frac{1}{\sqrt{2}}AB\]

Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circle is

- \[\sqrt{r}\]
- \[\sqrt{2}r AB\]
- \[\sqrt{3}r\]
- \[\frac{\sqrt{3}}{2}\]

If *AB* is a chord of a circle, *P* and *Q* are the two points on the circle different from *A* and *B*, then

∠

*APB*= ∠*AQB*∠

*APB*+ ∠*AQB*= 180° or ∠*APB*= ∠*AQB*∠

*APB*+ ∠*AQB*= 90°∠

*APB*+ ∠*AQB*= 180°

*AB* and *CD* are two parallel chords of a circle with centre *O* such that *AB* = 6 cm and *CD*= 12 cm. The chords are on the same side of the centre and the distance between them is 3 cm. The radius of the circle is

6 cm

- \[5\sqrt{2} cm\]
7 cm

- \[3\sqrt{5} cm\]

In a circle of radius 17 cm, two parallel chords are drawn on opposite side of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is

34 cm

15 cm

23 cm

30 cm

In the given figure, *O* is the centre of the circle such that ∠*AOC* = 130°, then ∠*ABC* =

130°

115°

65°

165°

## Chapter 15: Circles

## RD Sharma solutions for Mathematics for Class 9 chapter 15 - Circles

RD Sharma solutions for Mathematics for Class 9 chapter 15 (Circles) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics for Class 9 chapter 15 Circles are Angle Subtended by a Chord at a Point, Perpendicular from the Centre to a Chord, Equal Chords and Their Distances from the Centre, Angle Subtended by an Arc of a Circle, Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles, Cyclic Quadrilateral, Circles Passing Through One, Two, Three Points.

Using RD Sharma Class 9 solutions Circles exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 9 prefer RD Sharma Textbook Solutions to score more in exam.

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