CBSE Class 9 Mathematics Chapter 10: Circles NCERT Solutions
This resource provides comprehensive NCERT Solutions for Class 9 Mathematics, Chapter 10: Circles. It covers fundamental concepts related to circles, including their interior and exterior regions, the definition and properties of chords, semicircles, segments, and sectors. The solutions explain how a circle divides a plane and clarify the relationship between the radius and chords. It also delves into the properties of congruent circles and their chords, proving that equal chords subtend equal angles at the center and vice versa. These solutions are designed to help students understand the theoretical aspects of circles and prepare effectively for their examinations by reinforcing key definitions and theorems.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 9 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 10: Circles |
Chapter summary
Chapter 10, Circles, for Class 9 Mathematics NCERT Solutions focuses on the basic definitions and properties of circles. It includes exercises that test understanding of terms like interior, exterior, radius, diameter, chord, arc, segment, and sector. The chapter also proves theorems related to equal chords in congruent circles subtending equal angles at their centers. These solutions provide clear explanations and step-by-step reasoning for all questions.
Learning outcomes
- Understand the basic definitions related to a circle, including its center, radius, diameter, chord, arc, segment, and sector.
- Differentiate between the interior and exterior regions of a circle.
- Identify the longest chord of a circle.
- Define a semicircle and relate it to an arc.
- Explain how a circle divides a plane into three distinct parts.
- 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.
Topics covered
Paper topics
- Introduction to Circles
- 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
- Plane division by a circle
Important topics
- Definitions: Radius, Diameter, Chord, Arc, Segment, Sector
- Longest chord of a circle
- Number of parts a circle divides a plane into
- Congruence of circles
- Theorem: Equal chords subtend equal angles at the center
- Theorem: Chords subtending equal angles at the center are equal
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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 point equidistant from all points on the circumference. This point is 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 is greater than the radius, it means the point is further away from the center than any point on the circle's boundary. Therefore, the point lies in the exterior of the circle.
(iii) A chord is a line segment whose endpoints lie on the circle. The longest possible chord passes through the center of the circle, which is known as the diameter.
(iv) An arc is a portion of the circle's circumference. When the endpoints 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 region enclosed by an arc and the chord connecting the arc's endpoints. Thus, it is the region between an arc and the chord.
(vi) A circle divides the plane into three distinct regions: the area inside the circle (interior), the circle itself (the boundary), and the area outside the circle (exterior). Therefore, a circle divides the plane into three parts.
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 center of the circle to any point on its circumference.
(ii) False. A circle can have infinitely many chords of any given length. For any length less than or equal to the diameter, there are infinitely many chords of that length that can be drawn within the circle.
(iii) False. If a circle is divided into three equal arcs, each arc measures 360°/3 = 120°. An arc greater than a semicircle (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 passes through the center of the circle. By definition, a chord passing through the center is a diameter.
(v) False. A sector is the region bounded by two radii and the intercepted arc. The region between a chord and its corresponding arc is called a segment.
(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. Hence, it is a plane figure.
Question 1
Given: Two congruent circles with centres O and O'. Let AB and CD be equal chords of these circles, respectively. Since the circles are congruent, they have the same radius.
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, the triangles are congruent by the SSS (Side-Side-Side) congruence criterion.
Therefore, ΔAOB ≅ ΔCO'D.
By CPCT (Corresponding Parts of Congruent Triangles), the corresponding angles are equal.
Hence, ∠AOB = ∠CO'D. Proved.
Question 2
Given: Two congruent circles with centres O and O'. Let AB and CD be chords of these circles, respectively, such that the angles subtended by them at their centres are equal, i.e., ∠AOB = ∠CO'D. Since the circles are congruent, they have the same radius.
To Prove: The chords 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 at the centres 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, the triangles are congruent by the SAS (Side-Angle-Side) congruence criterion.
Therefore, ΔAOB ≅ ΔCO'D.
By CPCT (Corresponding Parts of Congruent Triangles), the corresponding sides are equal.
Hence, AB = CD. Proved.
Common mistakes
- Confusing major and minor arcs.
- Incorrectly identifying a sector as a segment or vice versa.
- Misunderstanding the relationship between a chord's length and the diameter.
- Errors in applying congruence criteria (SSS, SAS) in proofs involving circles.
Revision tips
- Memorize the definitions of all key terms related to circles.
- Draw diagrams for each question to visualize the geometric relationships.
- Focus on understanding the proofs for theorems involving congruent circles and their chords.
- Practice identifying different parts of a circle (radius, chord, arc, sector, segment) in given diagrams.
Practice MCQs
Q1. Where does the center of a circle lie?
Explanation: The center of a circle is defined as the point equidistant from all points on the circumference, and this point is located within the circle's boundary.
Q2. Which is the longest chord of a circle?
Explanation: The diameter is a chord that passes through the center of the circle and is twice the length of the radius, making it the longest possible chord.
Q3. A circle divides a plane into how many parts?
Explanation: A circle divides the plane into three regions: the interior of the circle, the circle itself (the boundary), and the exterior of the circle.
Q4. If two circles have the same radii, they are:
Explanation: Circles with equal radii are defined as congruent circles, meaning they have the same size and shape.
Q5. What is the region between an arc and the chord connecting its endpoints called?
Explanation: The region bounded by an arc and its corresponding chord is known as a segment of the circle.
Frequently asked questions
What are the key concepts covered in Chapter 10: Circles for Class 9 Maths?
Chapter 10 covers fundamental concepts of circles, including their interior and exterior regions, definitions of radius, diameter, chord, arc, segment, and sector. It also includes proofs related to equal chords in congruent circles.
How do these NCERT Solutions help in exam preparation?
These solutions provide clear, step-by-step explanations and proofs for all exercises in Chapter 10. They help students understand the theoretical aspects of circles and reinforce their knowledge for exams.
What is the difference between a segment and a sector of a circle?
A segment is the region between an arc and its corresponding chord. A sector is the region between an arc and the two radii connecting the center to the endpoints of the arc.
What is the longest chord of a circle?
The longest chord of a circle is its diameter. It is a chord that passes through the center of the circle.
How many parts does a circle divide a plane into?
A circle divides the plane into three parts: the interior region, the circle itself (the boundary), and the exterior region.
Are the proofs for congruent circles included in these solutions?
Yes, the solutions include proofs for the theorems stating that equal chords of congruent circles subtend equal angles at their centers, and vice versa.
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