CBSE Class 9 Maths Chapter 10 Circles NCERT Solutions
CBSE Class 9 Mathematics Chapter 10: Circles introduces students to the fundamental geometric properties of circles. This chapter explores concepts such as the interior and exterior of a circle, the diameter as the longest chord, and how a circle divides a plane into three parts. It also covers definitions of arcs, segments, and sectors. Key theorems are presented and proved, including the relationship between equal chords and the angles they subtend at the center, and vice versa. Understanding these properties is crucial for building a strong foundation in geometry and preparing effectively for examinations. The NCERT Solutions for this chapter offer clear, step-by-step explanations to help students grasp these concepts thoroughly.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 9 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 10 |
Chapter summary
Chapter 10, Circles, for Class 9 Maths NCERT Solutions, focuses on the basic definitions and properties of circles. It includes exercises on identifying parts of a circle like the interior, exterior, diameter, and semicircle. The chapter also covers concepts of arcs, segments, and sectors, and explores the relationship between equal chords and the angles they subtend at the center in congruent circles. These solutions provide step-by-step guidance for understanding and solving problems related to these geometric concepts.
Learning outcomes
- Understand the definitions of interior and exterior of a circle.
- Identify the diameter as the longest chord of a circle.
- Define and differentiate between arcs, segments, and sectors.
- Prove that equal chords of congruent circles subtend equal angles at their centers.
- Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
- Understand that a circle divides a plane into three distinct parts.
Topics covered
Paper topics
- Introduction to Circles
- Center of a Circle
- Interior and Exterior of a Circle
- Radius
- Diameter
- Chord
- Arc
- Semicircle
- Segment of a Circle
- Sector of a Circle
- Congruent Circles
- Angles subtended by chords at the center
Important topics
- Definitions of circle parts (radius, diameter, chord, arc, segment, sector)
- Properties of the longest chord (diameter)
- Relationship between equal chords and angles subtended at the center in congruent circles
- Understanding the three parts a circle divides a plane into
PDF preview
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Questions and Solutions
Question 1
(i) The centre of a circle lies in _____ of the circle. (exterior/interior)
(ii) A point, whose distance from the centre of a circle is greater than its radius, lies in _____ of the circle. (exterior/interior)
(iii) The longest chord of a circle is a _____ of the circle.
(iv) An arc is a _____ when its ends are the ends of a diameter.
(v) A segment of a circle is the region between an arc and _____ of the circle.
(vi) A circle divides the plane, on which it lies, into _____ parts.
(i) The centre of a circle is the fixed point from which all points on the circle are equidistant. This point is always located within the boundary of the circle, hence it lies in the interior of the circle.
(ii) If the distance of a point from the centre of a circle is greater than the radius, it means the point is further away from the centre than any point on the circle's circumference. Therefore, this point lies in the exterior of the circle.
(iii) A chord is a line segment connecting any two points on the circle. The longest possible chord passes through the centre of the circle, which is known as the diameter.
(iv) An arc is a portion of the circle's circumference. When the ends of an arc are the endpoints of a diameter, the arc forms exactly half of the circle's circumference, which is called a semicircle.
(v) A segment of a circle is defined as the area enclosed between a circular arc and the chord connecting the endpoints of that arc. So, it is the region between an arc and the chord.
(vi) A circle divides the plane it lies on into three distinct regions: the area inside the circle (the interior), the area outside the circle (the exterior), and the circle itself (the circumference).
Question 2
(i) A line segment joining the centre to any point on the circle is a radius of the circle.
(ii) A circle has only a finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
(v) A sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.
(i) True. By definition, a radius is a line segment connecting the centre of the circle to any point on its circumference.
(ii) False. A circle can have infinitely many equal chords. For any given chord length (less than or equal to the diameter), you can draw multiple chords of that same length at different positions within the circle.
(iii) False. If a circle is divided into three equal arcs, each arc measures 360°/3 = 120°. An arc measuring more than 180° is a major arc. Since 120° is less than 180°, each of these equal arcs is a minor arc.
(iv) True. A chord that is twice the length of the radius is equal to the diameter. The diameter is the longest chord of a circle and passes through the centre.
(v) False. The region between a chord and its corresponding arc is called a segment. A sector is the region bounded by two radii and the intercepted arc.
(vi) True. A circle is a two-dimensional shape drawn on a flat surface (a plane), consisting of all points equidistant from a central point. Therefore, it is a plane figure.
Question 1
Given: Two congruent circles with centres O and O'. Let AB and CD be two equal chords in these circles, respectively. This means AB = CD.
To Prove: The angles subtended by these equal chords at their respective centres are equal, i.e., ∠AOB = ∠CO'D.
Proof: Consider the triangles ΔAOB and ΔCO'D.
- AB = CD (Given that the chords are equal)
- AO = CO' (Radii of congruent circles are equal)
- BO = DO' (Radii of congruent circles are equal)
Since all three sides of ΔAOB are equal to the corresponding three sides of ΔCO'D (SSS congruence criterion), the two triangles are congruent.
\Delta AOB \cong \Delta CO'D
(By SSS axiom)
Corresponding Parts of Congruent Triangles are Equal (CPCT). Therefore, the angles subtended by these chords at the centres are equal:
\angle AOB = \angle CO'D
Hence Proved.
Question 2
Given: Two congruent circles with centres O and O'. Let AB and CD be chords in these circles such that the angles subtended at their centres are equal, i.e., ∠AOB = ∠CO'D.
To Prove: The chords AB and CD are equal, i.e., AB = CD.
Proof: Consider the triangles ΔAOB and ΔCO'D.
- AO = CO' (Radii of congruent circles are equal)
- BO = DO' (Radii of congruent circles are equal)
- ∠AOB = ∠CO'D (Given that the angles subtended are equal)
Since two sides and the included angle of ΔAOB are equal to the corresponding two sides and the included angle of ΔCO'D (SAS congruence criterion), the two triangles are congruent.
\Delta AOB \cong \Delta CO'D
(By SAS axiom)
Corresponding Parts of Congruent Triangles are Equal (CPCT). Therefore, the chords are equal:
AB = CD
Hence Proved.
Common mistakes
- Confusing major arcs with semicircles.
- Incorrectly identifying the region between a chord and its arc as a sector instead of a segment.
- Assuming a finite number of equal chords without considering the circle's size.
- Misinterpreting the definition of a sector.
Revision tips
- Clearly define all terms like radius, diameter, chord, arc, segment, and sector.
- Memorize the conditions for congruence of triangles (SSS, SAS) used in proving theorems.
- Practice drawing diagrams accurately for each problem involving circles.
- Review the reasons for True/False statements, especially those involving definitions and properties.
Practice MCQs
Q1. Where does the center of a circle lie?
Explanation: The center of a circle is defined as a point equidistant from all points on the circumference, and this point is always located within the circle's boundary.
Q2. What is the longest chord of a circle called?
Explanation: The diameter is a chord that passes through the center of the circle and is the longest possible line segment connecting two points on the circle's circumference.
Q3. A circle divides a plane into how many parts?
Explanation: A circle divides the plane into three distinct regions: the interior (where the circle's area lies), the exterior, and the circle itself (the boundary).
Q4. If two circles have the same radii, they are called:
Explanation: Congruent circles are identical in size and shape, meaning they have the same radius and can be superimposed on each other perfectly.
Q5. Which region is bounded by an arc and the two radii joining the center to the endpoints of the arc?
Explanation: This region is defined as a sector of the circle. A segment is bounded by an arc and a chord.
Frequently asked questions
What are the key concepts covered in CBSE Class 9 Maths Chapter 10: Circles?
This chapter covers the fundamental definitions and properties of circles, including the center, radius, diameter, chord, arc, segment, and sector. It also explores how circles divide a plane and the relationship between equal chords and the angles they subtend in congruent circles.
How do these NCERT Solutions help students?
These solutions provide clear, step-by-step explanations for each question in Chapter 10. They help students understand the concepts, verify their answers, and prepare effectively for their exams by reinforcing their knowledge of circle properties and theorems.
What is the difference between a segment and a sector of a circle?
A segment is the region bounded by an arc and its corresponding chord. A sector is the region bounded by an arc and the two radii joining the center to the endpoints of the arc.
Are the theorems in Exercise 10.2 important for exams?
Yes, the theorems in Exercise 10.2, which deal with the relationship between equal chords and the angles they subtend at the center in congruent circles, are fundamental and frequently tested. Understanding their proofs is crucial.
What does it mean for a circle to divide a plane into three parts?
A circle divides a plane into three distinct regions: the area inside the circle (interior), the area outside the circle (exterior), and the boundary of the circle itself.
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