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About Circles — Class 9 ICSE

Explore circle properties including chords, arcs, angles subtended, and cyclic quadrilaterals. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 12). On this page you can practice 50 questions across three difficulty levels — 10 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

  • Unit 1: Unveiling the Circle - Basic Definitions and Properties
  • Unit 2: Unlocking Chord Secrets - Properties and Theorems
  • Unit 3: Angles in a Circle - Arcs and Segments
  • Unit 4: Cyclic Quadrilaterals - Special Properties
  • Unit 5: Consolidating Circles - Review and Advanced Problem-Solving

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

Circles — solved examples for Class 9 ICSE

Example 1easy

In a circle with centre O, an arc ABC subtends an angle of 140° at the centre O. What is the measure of the angle ∠ADC, where D is any point on the remaining part of the circle?
  1. A)55°
  2. B)110°
  3. C)280°
  4. D)70°

Step-by-step solution

  1. The theorem states that 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. Here, arc ABC subtends ∠AOC = 140° at the centre O, and ∠ADC at point D on the remaining part.
  3. Therefore, ∠AOC = 2 × ∠ADC.
  4. Substituting the given value, 140° = 2 × ∠ADC, which means ∠ADC = 140° / 2 = 70°.

Answer: 70°

Example 2medium

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

Step-by-step solution

  1. Let the radius of the circle be r = 10 cm and the length of the chord be AB = 16 cm.
  2. The perpendicular from the center O to the chord AB bisects it. So, AM = MB = 16/2 = 8 cm.
  3. Consider the right-angled triangle OMA, where OA is the radius (hypotenuse), OM is the distance from the center, and AM is half the chord.
  4. Using Pythagoras theorem: OM² + AM² = OA². OM² + 8² = 10². OM² + 64 = 100. OM² = 36. So, OM = 6 cm.

Answer: 6 cm

Example 3hard

In a circle with centre O, chords AB and CD intersect at a point P inside the circle. If the chords make equal angles with the diameter passing through P, i.e., ∠APM = ∠DPM where PM is a segment of the diameter passing through P, which of the following statements must always be true?
  1. A)AB = CD
  2. B)AP = PD
  3. C)BP = PC
  4. D)AB || CD

Step-by-step solution

  1. Let OX be perpendicular to AB and OY be perpendicular to CD. Since the diameter passes through P and makes equal angles with AB and CD, the lines OX and OY must be equidistant from P along the diameter's perpendicular. Alternatively, consider the triangles formed by O, P, and the feet of the perpendiculars from O to AB and CD.
  2. In ΔOXP and ΔOYP, ∠OXP = ∠OYP = 90°. OP is common. We can show that ∠XOP = ∠YOP due to the equal angles made by the chords with the diameter (using alternate interior angles or other angle properties with the perpendiculars). Thus, ΔOXP ≅ ΔOYP by AAS congruence.
  3. From congruence, OX = OY. Since chords equidistant from the centre are equal in length, AB = CD.

Answer: AB = CD

Practice questions on Circles

  1. Q1.easy

    Points A, B, C, D lie on a circle in that order. Which of the following statements is always true?
    1. A)∠ABC = ∠ADC
    2. B)∠ABC + ∠ADC = 180°
    3. C)∠AOB = 2 × ∠ADB (O is the centre)
    4. D)∠BAC = ∠BDC
    Show answer

    Answer: ∠ABC + ∠ADC = 180°

    Hint: Consider the properties of a quadrilateral whose vertices lie on a circle.

  2. Q2.easy

    If AB is the diameter of a circle and C is a point on the circumference, then the angle ∠ACB is always _______.
    1. A)acute
    2. B)obtuse
    3. C)a right angle
    4. D)variable
    Show answer

    Answer: a right angle

    Hint: A diameter divides a circle into two semi-circles. Consider the arc that subtends an angle at the circumference when the chord is a diameter.

  3. Q3.easy

    A quadrilateral ABCD is inscribed in a circle. If ∠A = 75° and ∠B = 100°, what are the measures of ∠C and ∠D respectively?
    1. A)∠C = 80°, ∠D = 105°
    2. B)∠C = 75°, ∠D = 100°
    3. C)∠C = 105°, ∠D = 100°
    4. D)∠C = 105°, ∠D = 80°
    Show answer

    Answer: ∠C = 105°, ∠D = 80°

    Hint: Recall the property of opposite angles in a cyclic quadrilateral.

  4. Q4.medium

    Which of the following statements is TRUE regarding a circle?
    1. A)The longest chord of a circle is called its radius.
    2. B)A segment of a circle is the region between an arc and two radii.
    3. C)Equal chords of a circle subtend equal angles at the center.
    4. D)The angle subtended by an arc at the center is half the angle subtended by it at any point on the remaining part of the circle.
    Show answer

    Answer: Equal chords of a circle subtend equal angles at the center.

    Hint: Carefully recall the definitions of parts of a circle and the theorems related to chords and angles.

  5. Q5.medium

    In a circle with center O, an arc AB subtends an angle of 70° at the center. What is the measure of the angle subtended by the same arc AB at a point C on the remaining part of the circle?
    1. A)35°
    2. B)70°
    3. C)140°
    4. D)210°
    Show answer

    Answer: 35°

    Hint: Consider the relationship between the angle subtended by an arc at the center and the angle subtended by the same arc at any point on the circumference.

  6. Q6.medium

    Two chords AB and CD of lengths 12 cm and 16 cm respectively are parallel to each other and are on opposite sides of the center of a circle. If the radius of the circle is 10 cm, find the distance between the two chords.
    1. A)14 cm
    2. B)10 cm
    3. C)8 cm
    4. D)7 cm
    Show answer

    Answer: 14 cm

    Hint: Draw perpendiculars from the center to both chords. Use the Pythagorean theorem for each chord to find their individual distances from the center, then add them.

  7. Q7.hard

    A circle has a radius of 10 cm. Two parallel chords, AB and CD, are of lengths 16 cm and 12 cm respectively. If both chords lie on the same side of the centre, what is the distance between them?
    1. A)2 cm
    2. B)3 cm
    3. C)4 cm
    4. D)5 cm
    Show answer

    Answer: 2 cm

    Hint: Use the property that the perpendicular from the centre to a chord bisects the chord. Apply Pythagoras theorem twice to find the distance of each chord from the centre.

  8. Q8.hard

    In a circle with centre O, points A, B, C are on the circumference. If ∠BOC = 120° and ∠AOB = 90°, what is the measure of ∠ACB?
    1. A)45°
    2. B)60°
    3. C)75°
    4. D)105°
    Show answer

    Answer: 75°

    Hint: First, find the angle subtended by arc AC at the centre. Remember that angles around a point sum to 360°, and the angle at the centre is double the angle at the circumference.

  9. Q9.hard

    Ravi was asked to find the length of a chord in a circle with radius 13 cm, given that its distance from the centre is 5 cm. He reasoned: "Since the perpendicular from the centre bisects the chord, and the radius forms a right-angled triangle with half the chord and the distance from the centre, then the chord length is 2 × √(13² + 5²) = 2 × √(169 + 25) = 2 × √194 cm." What mistake did Ravi make?
    1. A)He incorrectly identified the hypotenuse of the right-angled triangle.
    2. B)He used the wrong formula for the perpendicular from the centre to a chord.
    3. C)He calculated the square root incorrectly.
    4. D)He should have used the formula 2 × √(13² - 5²) instead.
    Show answer

    Answer: He incorrectly identified the hypotenuse of the right-angled triangle.

    Hint: In the right-angled triangle formed, the radius is always the hypotenuse because it connects the centre to a point on the circumference.

These are 9 of the 50 questions available for Circles. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.