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NCERT Solutions for Class 9 Maths Chapter 9 Circles

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NCERT Solutions for Class 9 Chapter 9 Circles Maths - FREE PDF Download

Circles Class 9 NCERT Solutions, provide comprehensive guidance on understanding the properties and theorems related to circles. This chapter is crucial as it lays the foundation for concepts like tangents, chords, and the different segments of a circle.

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These solutions help students to grasp the key points and solve complex problems with ease. Focus areas include practising theorems, understanding their proofs, and applying these concepts to different problems. The Class 9 Maths NCERT Solutions detailed explanations and step-by-step approach make it easier for students to prepare effectively for their exams and build a strong mathematical base.


Glance of NCERT Solutions for Class 9 Maths Chapter 9 Circles | Vedantu

  • In Chapter 9 Class 9 Maths, focused on circles, several fundamental concepts and terminologies are essential for understanding this key geometric shape.

  • A circle is defined by all the points in a plane that lie at a fixed distance, known as the radius, from a central point.

  • The diameter of a circle is a special type of chord—a line segment whose endpoints are on the circle—that passes through the circle's centre and is twice the length of the radius.

  • An arc represents a portion of the circle’s circumference, while a sector is the area enclosed by two radii and the arc between them. 

  • A segment, on the other hand, is the area bounded by a chord and the arc subtended by that chord. 

  • This chapter delves into the properties of circles, demonstrating that a line bisects and is perpendicular to a chord, equal chords are equidistant from the centre.

  • This chapter explains about circles' theorems, focusing on cyclic quadrilateral angles and the principle that an angle subtended by an arc at the centre is twice that subtended at any other point.

  • This article contains chapter notes, important questions, exemplar solutions, exercises, and video links for Chapter 9 - Circles, which you can download as PDFs.

  • There are Exercise links provided. It has solutions for each question from Circles.

  • There are three exercises (20 fully solved questions) in Chapter 9 Class 9 Maths Circles.


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Exercises under NCERT Solutions for Class 9 Maths Chapter 9 Circles

NCERT Solutions for Class 9 Maths Chapter 9, "Circles," contains six exercises that cover different topics related to circles. Here is a brief overview of the types of questions dealt with in each exercise:


  • Exercise 9.1: This exercise consists of two questions that are based on the tangents and the properties of tangents of a circle. The questions include finding the length of the tangent, the angle between the tangent and the radius, and the distance of the point from the centre of the circle.

  • Exercise 9.2: This exercise consists of three questions that are based on the secants of a circle. The questions include finding the length of the secant, the intersection point of two secants, and the angle between the secant and the tangent.

  • Exercise 9.3: This exercise consists of six questions that are based on the chords of a circle. The questions include finding the length of the chord, the angle between the chords, and the perpendicular bisector of the chord.


Access NCERT Solutions for Class 9 Maths Chapter 9 – Circles

Exercise 9.1

1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Ans:

As we know that a circle is a collection of points therefore, they are equidistant from a fixed point. Now, this fixed point will be the centre of the circle and the equal distance between these points will be the radius of the circle. Hence, the shape of a circle will depend on its radius. Therefore, when we superimpose two circles of equal radius, then both the circles will cover each other. Thus, these two circles will be congruent when they have equal radius. Now, let us assume that two congruent circles have a common centre: O and O, AB and CD are the two chords of same length.


two circles with the same radii


In ΔAOB and ΔCOD, we can observe that 

AB=CD as they are chords of the same length. 

OA=OC as they are radii of congruent circles, 

OB=OD as they are radii of congruent circles.

Therefore, ΔAOBΔCOD by the SSS congruence rule. This implies AOBCOD By CPCT. Hence, equal chords of congruent circles subtend equal angles at their centres.


2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Ans:

Let us assume that there are two congruent circles with the same radii that have centres as O and O.


chords of congruent circles subtend equal angles at their centres, then the chords are equal.


In ΔAOB and ΔCOD

AOB=COD (Given) 

OA=OC as they are radii of congruent circles 

OB=OD as they are radii of congruent circles 

Therefore,

ΔAOBΔCOD by the SSS congruence rule. 

AB=CD (By CPCT) 

Hence, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.


Exercise 9.2

1. Two circles of radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of the common chord.

Ans: Let us assume that the radius of the circle which is centred at O and O be 5cm and 3cm.


Two circles of radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm . Find the length of the common chord


Therefore,

OA=OB

5cm

Similarly,

OA=OB

3cm

Now, the line segment OO will be the perpendicular bisector of the chord AB.True. 

We know that the points on the circle are always on equal distances from the centre of the circle and hence, this equal distance is defined as the radius of the circle. This is why a line segment joining the centre to any point on the circle is a radius of the circle.


2. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Ans:

Let us assume that PQ and RS are the two chords of equal length of a circle and they are intersecting at a common point T.


PQ and RS are the two chords of equal length of a circle and they are intersecting at a common point T


So, let us draw two perpendicular bisectors OV and OU on these chords.

In ΔOVT and ΔOUT,

We have OV=OU as they are equal chords of a circle and are equidistant from the centre.

Also, OVT=OUT.

Therefore,

ΔOVTΔOUT, by the RHS congruent rule.

VT=UT by CPCT.

Now, we have given that –

PQ=RS

12PQ=12RS

PV=RU.

Now, let us add both the conditions as –

PV+VT=RU+UT

PT=RT.

On subtracting we get –

PQPT=RSRT

This equation indicates that a corresponding segment of the chords are congruent to each other. Hence, proved.


3. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Ans:

Let us assume that PQ and RS are the two chords of the same length of a circle which are intersecting at a common point T.


the line joining the point of intersection to the centre makes equal angles with the chords


So, let us draw two perpendicular bisectors OV and OU on these chords.

In ΔOVT and ΔOUT,

We have OV=OU as they are equal chords of a circle and are equidistant from the centre.

Also, OVT=OUT.

Therefore,

ΔOVTΔOUT, by the RHS congruence rule.

Therefore, we can conclude that OVT=OUT by CPCT. Hence, if two equal chords of a circle intersect within the circle, then the line joining the point of intersection to the centre makes equal angles with the chords. Hence, proved.


4. If a line intersects two concentric circles (circles with the same centre) with centre O at A,B,C and D, prove that AB=CD.


AB=CD


Ans:

In the figure, let us draw a perpendicular OM bisecting the chord BC and AD.


a perpendicular OM bisecting the chord BC and AD


We can observe from the figure that BC<AD.

Hence, we have –

BM=MC and

AM=MD.

On subtracting both equations, we get –

AMBM=MDMC

AB=CD.

Hence, proved.


5. Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma and Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Ans:

Let us assume that OA and OB are the two perpendiculars of RS and SM as shown in the figure below.


OA   and OB are the two perpendiculars of RS and SM


Hence, we have –

AR=AS

3m.

Also, OR=OS=OM=5m.

In ΔOAR,

OA2+AR2=OR2

OA2=259

OA=4m.

As, from the figure we can observe that ORSM is a kite. Now, we know that the diagonals of a kite are perpendicular.

Therefore,

RCS=90 and RC=CM.

Area of the ΔORS=12×OA×RS

12×RC×OS=12×4×6

RC=4.8

Hence,

RM=2RC

RM=9.6m.

Therefore, the distance between Reshma and Mandip will be 9.6m.


6. A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

Ans:

Let us draw a figure as –


the length of the string of each phone


From the figure, we can observe that AS=SD=DA.

Hence, ΔASD will be an equilateral triangle and OA=20m.

Now, we know that the medians of an equilateral triangle will pass through the centre. Also, the medians will intersect each other at the ratio 2:1.

Therefore, the median AB is –

OAOB=21

20OB=21

OB=10m

Hence, AB=OA+OB

AB=30m.

In ΔABD, we have –

AD2=AB2+BD2

AD2=900+(AD2)2

3AD2=3600

AD=203

Hence, the length of the string of each phone will be 203m.


Exercise 9.3

1. In the given figure, A,B, and C are three points on a circle with centre O such that BOC=30 and AOB=60. If D is a point on the circle other than the arc ABC, find ADC.


∠AOC=∠AOB+∠BOC


Ans:

From the figure, we can observe that –

AOC=AOB+BOC

AOC=60+30

AOC=90.

As, the angle subtended by the arc at the centre will be twice the angle on the remaining part. Therefore,

ADC=12(90)

ADC=45.


2. 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.

Ans:

In ΔOAB,


the chord at a point on the minor arc and also at a point on the major arc


We have –

AB=OA=OB as radius.

Hence, ΔOAB will be an equilateral triangle.

This implies that each interior angle of the equilateral triangle will be 60.

AOB=60

ACB=12AOB

12(60)=30.

In quadrilateral ACBD,

We have –

ACB+ADB=180

ADB=150.

Therefore, the angle subtended by the chord on the major and minor arc will be 30 and 150.


3. In the given figure, PQR=100, where P,Q, and R are points on a circle with centre O. Find OPR.


∠PQR=100∘  , where P,Q, and R are points on a circle with centre O


Ans:

Let us assume that PR is a chord of the circle and S is any point on the major arc.


PR is a chord of the circle and S is any point on the major arc


PQRS is a cyclic quadrilateral.

Hence, we have –

PQR+PSR=180

PSR=80

Now, we know that the angle subtended by the arc at centre will be double the angle subtended by it.

Therefore,

PQR=2PSR

POR=160

In ΔPOR,

We can observe that –

OP=PR.

OPR=ORP as they are opposite angles of equal sides of a triangle.

OPR+ORP+POR=180 which is the angle sum property of a triangle.

2OPR+160=180

OPR=10

Therefore, OPR=10.


4. In figure, ABC=69, ACB=31, find BDC?


∠ABC=69* ,∠ACB=31*


Ans:

From the given figure, we have –

BAC=BDC.

In ΔABC,

BAC+ABC+ACB=180

BAC=1806931

BAC=80.

Therefore, we have BDC=80.


5. In the given figure, A,B,C and D are four points on a circle. AC and BD intersect at a point E such that BEC=130 and ECD=20. Find BAC.


A,B,C and D are four points on a circle


Ans:

From the given figure, we have –

In ΔCDE,

CDE+DCE=CEB

CDE=13020

CDE=110.

But we know that CDE=BAC

Therefore,

BAC=110.


6. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If DBC=70, BAC=30, find BCD. Further, if AB=BC, find ECD.

Ans:

The figure will be as –


ABCD   is a cyclic quadrilateral whose diagonals intersect at a point E


From figure, we can observe that –

CBD=CAD

CAD=70.

BAD=BAC+CAD

BAD=100

Therefore, we have –

BCD+BAD=180

BCD=80.

Now, in ΔABC, we have –

AB=BC

BCA=CAB

BCA=30.

Also, we have –

BCD=80

BCA+ACD=80

ACD=8030

ACD=50

ECD=50.


7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Ans:

Let us assume a cyclic quadrilateral ABCD having diagonals BD and AC, intersecting at a common point O.


a cyclic quadrilateral ABCD  having diagonals BD  and AC


BAD=12BOD

BAD=90

Now, BCD+BAD=180

BCD=90.

ADC=12AOC

ADC=90

ADC+ABC=180

ABC=90.

Therefore, each interior angle of the quadrilateral is 90 which implies that ABCD is a rectangle.


8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Ans:

Let us assume a trapezium ABCD with ABCD and BC=AD as shown in the figure below.


A trapezium ABCD  with AB∥CD and BC=AD


From the figure, we can observe that AMCD and BNCD.

Therefore, in ΔAMD and ΔBNC,

AD=BC.

AMD=BNC

AM=BN

ΔAMDΔBNC by the RHS congruence rule.

Therefore, ADC=BCD.

BAD and ADC are on the same side.

Therefore, BAD+ADC=180

BAD+BCD=180

Hence, the angles are supplementary. Therefore, ABCD is a cyclic quadrilateral.


9. Two circles intersect at two points B and C. Through B, two-line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ACP=QCD.


Two circles intersect at two points B  and C


Ans:

Let us join the chords AP and DQ.

Therefore,

PBA=ACP,

Also, DBQ=QCD.

Now, we know that ABD and PBQ are the line segments intersecting at common point B.

Therefore,

PBA=DBQ

Hence, we have ACP=QCD.


10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Ans:

Let us consider a triangle ΔABC in the figure given below –


a triangle ΔABC


We can observe that two circles are drawn by taking the diameters AB and AC. We will let the points B and C intersect each other at a common point D which does not lie on the line segment BC.

Therefore, after joining AD we have –

ADB=90

ADC=90

BDC=ADB+ADC

BDC=180.

Hence, we have a straight line as BDC. This implies that the assumption that we considered was wrong.

Therefore, the point of intersection D will lie on the line segment BC.


he point of intersection D will lie on the line segment BC


11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that CAD=CBD.

Ans:

From the figure we know that in ΔABC,


ABC   and ADC  are two right triangles with common hypotenuse AC


ABC+BCA+CAB=180

BCA+CAB=90.

In ΔADC,

CDA+ACD+DAC=180

ACD+DAC=90

After adding both the conditions, we get –

BCA+CAB+ACD+DAC=180

(BCA+ACD)+(CAB+DAC)=180

BCD+DAB=180.

Now, we know that B+D=180.

Therefore, we can observe from the sum of each interior angle that it is a cyclic quadrilateral.


ΔABC


Hence,

CAD=CBD.


12. Prove that a cyclic parallelogram is a rectangle.

Ans:

Let us assume a cyclic parallelogram ABCD as shown in the figure below –


a cyclic parallelogram is a rectangle


We have –

A+C=180.

Now, we know that in a parallelogram opposite angles are always equal.

Therefore,

A=C and

B=D.

A+C=180

A=90.

Similarly,

B=90.

Therefore, all the interior angles of the parallelogram are 90 which implies it is a rectangle. Hence, proved.


Overview of Deleted Syllabus for CBSE Class 9 Maths Circles

Chapter - 9

Dropped Topics

Circles

Exercise 9.1 Introduction

Exercise 9.2 Circles and its related terms: Review and Circle through three points.


Class 9 Maths Chapter 9: Exercises Breakdown

Chapter 9 -  Circles Exercises in PDF Format

Exercise 9.1

2 Questions with Solutions

Exercise 9.2

6 Questions with Solutions

Exercise 9.3 

12 Questions with Solutions


Conclusion 

NCERT Chapter 9 of Class 9 Maths, "Circles," covers essential concepts and theorems related to circles. Key topics include properties of chords, angles subtended by chords, and cyclic quadrilaterals. Important theorems to focus on are those involving the perpendicular from the centre of a chord, equal chords and their distances from the centre, and the unique circle passing through three non-collinear points. Understanding these concepts and practising the related exercises can enhance logical thinking and problem-solving skills. NCERT solutions offer detailed explanations and a variety of solved and unsolved problems, which are crucial for thorough preparation.


Other Study Material for CBSE Class 9 Maths Chapter 9


Chapter-Specific NCERT Solutions for Class 9 Maths

Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.



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FAQs on NCERT Solutions for Class 9 Maths Chapter 9 Circles

1. Define Sectors?

The sector is an interior part of a circle which is covered by two radii and one arc of that Circle. These are of two types. The minor sector, which covers less area. The major sector, which covers a large area.

2. Distinguish Between Chord and Diameter?

A chord is a line segment drawn between any two points in the circle. At the same time, the diameter is drawn in the middle of the circle. It cuts the circle into two halves. The radius of a circle is defined as a half part of the diameter.


d = 2r.

3. How are NCERT Solutions for Class 9 Maths Chapter 9 helpful for Class 9 students?

Students who aim to excel in their CBSE Board exams have NCERT textbooks as the best study material at their disposal. In order to fully comprehend the solutions of NCERT textbook problems, students can refer to Vedantu's NCERT Solutions for Class 9 Maths Chapter 9 and download it for free. These solutions offer on point solutions prepared by experts that can help you ace the material in no time. 

4. Why should we follow NCERT Solutions for Class 9 Maths Chapter 9?

Here is why you should follow Vedantu's NCERT Solutions:

  • If you are attempting CBSE Board exams, then a rigorous study of the NCERT syllabus should be your first most priority. 

  • It is equally important to practice the NCERT problems and properly understand their solutions.

  • Vedantu's NCERT Solutions for Class 9 Maths are curated by Mathematics experts.

  • The solutions are offered in a direct, logical and understandable way making the concepts easy for students to grasp. 

5. What are the basics of Class 9 Maths Chapter 9 Circles?

Chapter 9 Circles talks about the basic concepts and terms related to a circle. The chapter has some important theorems for students to learn. Basic concepts of the chapter include:

  • Angle Subtended by the Chord at a Point

  • Perpendicular From the Centre of a Chord

  • Circles Through Three Points

  • Chords and Their Distance From the Centre

  • Angle Subtended by the Area of Circle

  • Cyclic Quadrilaterals

6. How do I solve the problems of Class 9 Maths Chapter 9 Circles?

Before attempting to solve any problems, it is important that the core concepts are clear in a student’s mind. In order to correctly solve the problems and understand the concepts, students can refer to NCERT Solutions for Class 9 Maths Chapter 9 Circles. These solutions can also be accessed from the Vedantu app. These solutions attempt to solve the problems in the most logical and straightforward way as prepared by Vedantu’s Mathematics expert teachers. 

7. What are the most important theorems that come in Class 9 Chapter 9 Circles?

Class 9 Maths Chapter 9 Circles has a lot of theorems. All the theorems mentioned in the chapter are important, and students should practice them well. The main theorems include:

  • Equal chords of a circle subtend equal angles at the centre.

  • The perpendicular from the centre of a circle to a chord bisects the chord

  • Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).

In addition to these, all theorems are important and must be practised well. 

8. What is the most important theorem in circles for Class 9?

The most important theorem in Class 9 circles is Thales' Theorem (also known as the Angle in a Semi-Circle Theorem). Thales' Theorem is a key concept in Class 9 circles, stating that if you draw a straight line (a diameter) across the middle of a circle and then make an angle from any point on the circle's edge, that angle will always be 90 degrees, or a right angle. Essentially, this theorem helps to explain why any triangle formed with the endpoints of the diameter and a point on the circle is always a right triangle, showcasing a fundamental property of circles in geometry.

9. What is special about circles in Circles Class 9?

In Circles Class 9 , circles are special because they introduce fundamental concepts and properties such as:

  • Chord properties: Equal chords are equidistant from the centre.

  • Angle properties: Angles subtended by the same arc are equal, and the angle in a semicircle is a right angle.

  • Tangent properties: A tangent to a circle is perpendicular to the radius at the point of tangency.

10. How to prove a circle in Ch 9 Maths Class 9?

To prove a circle in Ch 9 Maths Class 9, you can demonstrate that all points on the circle are equidistant from a fixed point called the centre. This involves showing that every line segment drawn from the centre to the circumference (the radius) is of equal length. Additionally, you can verify that any point on the circumference maintains a constant distance from the centre, ensuring that this distance remains the same for all points around the circle. These properties confirm that the shape in question is indeed a circle, as they align with the fundamental definition of a circle in geometry.

11. How many types of circles are there in Ch 9 Maths Class 9?

While there is fundamentally one type of geometric figure called a circle, circles can be categorized based on their relative positions and relationships with each other:

  • Concentric Circles: Circles that share the same centre but have different radii.

  • Tangent Circles: Circles that touch each other at exactly one point. They can be internally tangent (one circle inside another) or externally tangent (touching from outside).

  • Congruent Circles: Circles with the same radius.

12. What is a circle in Class 9 Maths Ch 9?

A circle is essentially a shape where every point along its edge is the exact same distance from a central point. This constant distance is known as the radius. Imagine drawing a perfect loop around a point, making sure every part of the loop is equally far from the centre — that's your circle.

13. What is a chord in a circle Class 9 Maths Ch 9?

A chord in a circle is like a straight road that cuts straight across the circle, touching it at two points on the perimeter. This line doesn't go all the way around; instead, it just stretches from one side of the circle to the other, connecting those two points directly.