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About Circles — Class 9 CBSE

Study chords, arcs, angles subtended by chords, and properties of cyclic quadrilaterals. This topic is part of the CBSE Class 9 mathematics syllabus (chapter: Chapter 10). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Circles

  • Introduction to Circles and Their Parts
  • Chord Properties of a Circle
  • Angles Subtended by an Arc
  • Cyclic Quadrilaterals
  • Summary, Connections, and Practice Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Circles — solved examples for Class 9 CBSE

Example 1easy

Which of the following statements correctly defines a 'segment' of a circle?
  1. A)A) The region between an arc and two radii joining the center to the endpoints of the arc.
  2. B)B) The region between a chord and its corresponding arc.
  3. C)C) The entire boundary of the circle.
  4. D)D) A line segment connecting two points on the circle.

Step-by-step solution

  1. A segment of a circle is defined as the region bounded by a chord and its corresponding arc.
  2. Option A describes a sector, option C describes the circumference, and option D describes a chord.

Answer: B) The region between a chord and its corresponding arc.

Example 2medium

Which of the following statements about a circle is TRUE?
  1. A)A segment is the region between a chord and an arc.
  2. B)A sector is the region between two radii and a chord.
  3. C)A chord is always longer than the diameter.
  4. D)The longest chord of a circle is called its radius.

Step-by-step solution

  1. A segment is defined as the region bounded by a chord and either of its arcs.
  2. A sector is the region bounded by two radii and the arc between them, not a chord.
  3. A chord can be shorter than, equal to, or longer than the diameter; however, the longest chord is the diameter itself, meaning a chord is never longer than the diameter.
  4. The longest chord of a circle is its diameter, not its radius.

Answer: A segment is the region between a chord and an arc.

Example 3hard

Two parallel chords of a circle, of lengths 10 cm and 24 cm, are on opposite sides of the centre. If the radius of the circle is 13 cm, what is the distance between the chords?
  1. A)10 cm
  2. B)12 cm
  3. C)17 cm
  4. D)20 cm

Step-by-step solution

  1. Let the radius of the circle be r = 13 cm. For the chord of length 10 cm, its half-length is 10/2 = 5 cm. Let its perpendicular distance from the centre be d1.
  2. Using Pythagoras theorem: d1² + 5² = 13² => d1² + 25 = 169 => d1² = 144 => d1 = 12 cm.
  3. For the chord of length 24 cm, its half-length is 24/2 = 12 cm. Let its perpendicular distance from the centre be d2.
  4. Using Pythagoras theorem: d2² + 12² = 13² => d2² + 144 = 169 => d2² = 25 => d2 = 5 cm. Since the chords are on opposite sides of the centre, the distance between them is d1 + d2 = 12 cm + 5 cm = 17 cm.

Answer: 17 cm

Practice questions on Circles

  1. Q1.easy

    In a circle, if two chords AB and CD are equal in length, which of the following statements must be true?
    1. A)A) They are parallel to each other.
    2. B)B) They subtend equal angles at the center of the circle.
    3. C)C) They are equidistant from the center.
    4. D)D) Both B and C.
    Show answer

    Answer: D) Both B and C.

    Hint: Remember the theorems relating chord length to the angle subtended at the center and the distance from the center.

  2. Q2.easy

    A chord of length 16 cm is drawn in a circle with a radius of 10 cm. What is the distance of the chord from the center of the circle?
    1. A)A) 6 cm
    2. B)B) 8 cm
    3. C)C) 10 cm
    4. D)D) 12 cm
    Show answer

    Answer: A) 6 cm

    Hint: Draw a diagram. The perpendicular from the center to the chord bisects the chord, forming a right-angled triangle.

  3. Q3.easy

    Ravi was asked to prove that if a line segment from the center of a circle bisects a chord, then it is perpendicular to the chord. He started by assuming the line is perpendicular and then proved it bisects the chord. Where did Ravi make a mistake in his reasoning?
    1. A)A) He used the Pythagorean theorem incorrectly.
    2. B)B) He assumed what he needed to prove (circular reasoning).
    3. C)C) He did not draw a diagram.
    4. D)D) He should have used similar triangles.
    Show answer

    Answer: B) He assumed what he needed to prove (circular reasoning).

    Hint: In a proof, you must start with the given conditions and logically derive the conclusion. You cannot assume the conclusion.

  4. Q4.medium

    A chord of a circle is 16 cm long and is at a distance of 6 cm from the centre. What is the radius of the circle?
    1. A)8 cm
    2. B)10 cm
    3. C)12 cm
    4. D)14 cm
    Show answer

    Answer: 10 cm

    Hint: Remember that the perpendicular from the centre to a chord bisects the chord. This forms a right-angled triangle.

  5. Q5.medium

    In a circle with centre O and radius 5 cm, a chord AB is drawn such that the distance of the chord from the centre is 3 cm. What is the length of the chord AB?
    1. A)4 cm
    2. B)6 cm
    3. C)8 cm
    4. D)10 cm
    Show answer

    Answer: 8 cm

    Hint: Draw a perpendicular from the centre to the chord. This forms a right-angled triangle where the radius is the hypotenuse.

  6. Q6.medium

    Ravi states that if two chords of a circle are equal in length, then they must be at the same distance from the centre. Which theorem supports Ravi's statement?
    1. A)The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
    2. B)The perpendicular from the centre of a circle to a chord bisects the chord.
    3. C)Equal chords of a circle are equidistant from the centre.
    4. D)Angles in the same segment of a circle are equal.
    Show answer

    Answer: Equal chords of a circle are equidistant from the centre.

    Hint: Consider the relationship between chord length and its distance from the centre. One of the fundamental theorems directly links these two properties.

  7. Q7.hard

    Ravi states: 'If two chords are equidistant from the centre of a circle, then they must be equal in length.' This statement is the converse of the theorem 'If two chords of a circle are equal, then their perpendicular distances from the centre are equal.' Which of the following best describes the validity of Ravi's converse statement?
    1. A)It is always true.
    2. B)It is sometimes true, depending on the circle's radius.
    3. C)It is false, as chords can be equidistant but different lengths.
    4. D)It is true only if the chords are parallel.
    Show answer

    Answer: It is always true.

    Hint: Consider the proof for this theorem. If the perpendicular distances from the centre are equal, what does that imply about the half-lengths of the chords when using the Pythagorean theorem?

  8. Q8.hard

    In a circle with center O, A, B, C are three distinct points on the circumference. If ∠AOC = 140°, and B is a point on the minor arc AC, then what is the measure of ∠ABC?
    1. A)70°
    2. B)110°
    3. C)140°
    4. D)220°
    Show answer

    Answer: 110°

    Hint: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Remember to consider which arc subtends ∠ABC.

  9. Q9.hard

    In a circle, AB is a diameter. C and D are two points on the same semi-circle. If ∠CAB = 40° and ∠ABD = 30°, what is ∠CAD?
    1. A)40°
    2. B)30°
    3. C)20°
    4. D)10°
    Show answer

    Answer: 20°

    Hint: An angle inscribed in a semicircle is a right angle. Use this property to find other angles in the triangles formed.

These are 9 of the 60 questions available for Circles. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.