Question Bank — Circles

NCERT Class 10 Mathematics · Chapter 10 · CBSE Board Pattern

Class 10 Mathematics Chapter 10 Question Bank CBSE Board Pattern

Section A — MCQs (10 questions, 1 mark each)

  1. A line that intersects a circle at exactly one point is called:
    (A) Secant (B) Tangent (C) Chord (D) Radius

  2. The common point of a tangent and the circle is called:
    (A) Centre (B) Point of contact (C) Mid-point (D) Focus

  3. How many tangents can be drawn to a circle from a point lying inside the circle?
    (A) 0 (B) 1 (C) 2 (D) Infinite

  4. The tangent at any point of a circle is perpendicular to the:
    (A) Chord (B) Diameter (C) Radius through the point of contact (D) Secant

  5. From an external point, the number of tangents that can be drawn to a circle is:
    (A) 0 (B) 1 (C) 2 (D) 3

Assertion-Reason Questions

  1. Assertion (A): The length of tangents drawn from an external point to a circle are equal.
    Reason (R): The two tangents from an external point subtend equal angles at the centre.
    (A) Both A and R are true and R is the correct explanation of A.
    (B) Both A and R are true but R is not the correct explanation of A.
    (C) A is true but R is false.
    (D) A is false but R is true.

  2. Assertion (A): At any point on a circle, there can be only one tangent.
    Reason (R): The tangent is the limiting case of a secant when the two points of intersection coincide.
    (A) Both A and R are true and R is the correct explanation of A.
    (B) Both A and R are true but R is not the correct explanation of A.
    (C) A is true but R is false.
    (D) A is false but R is true.

  3. In two concentric circles, a chord of the larger circle touching the smaller circle is:
    (A) Bisected at the point of contact (B) Not bisected (C) Equal to radius (D) Parallel to diameter

  4. If two tangents drawn from an external point are inclined at 70° to each other, then the angle between the line joining the centre and the external point is:
    (A) 70° (B) 110° (C) 140° (D) 90°

  5. The line containing the radius through the point of contact is also called:
    (A) Secant (B) Normal (C) Chord (D) Diameter

Section B — Very Short Answer (6 questions, 2 marks each)

  1. State the number of tangents that can be drawn to a circle from a point (i) inside the circle, (ii) on the circle, (iii) outside the circle.

  2. Fill in the blanks:
    (i) A tangent to a circle intersects it in __ point(s).
    (ii) A circle can have     ____ parallel tangents at the most.

  3. Define the length of the tangent from an external point to a circle.

  4. Why can no tangent be drawn from a point inside the circle?

  5. What is the relationship between a tangent and the radius through the point of contact?

  6. State Theorem 10.2 in your own words.

Section C — Short Answer (5 questions, 3 marks each)

  1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle.

  2. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

  3. The length of a tangent from a point A at a distance of 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

  4. In the given figure, if TP and TQ are two tangents to a circle with centre O such that ∠POQ = 110°, find ∠PTQ.

  5. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Section D — Long Answer (5 marks each)

  1. Prove that the lengths of tangents drawn from an external point to a circle are equal. (Theorem 10.2)

  2. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. Show all steps.

  3. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC. (Multi-step problem)

Section E — Case/Source-Based (4 marks each)

Case 1: Bicycle Wheel

A bicycle wheel moves on the ground. The spokes are radii and the ground acts as a tangent at the point of contact.

(i) What is the relationship between the radius and the tangent at the point of contact?
(ii) How many tangents can be drawn from the centre of the wheel to the point of contact?
(iii) If the radius of the wheel is 35 cm, what is the angle between the radius and the ground?
(iv) Explain why the wheel moves along a tangent.

Case 2: Pulley System

A pulley used to draw water from a well has a rope on both sides acting as tangents to the circular pulley.

(i) From an external point (where the rope leaves the pulley), how many tangents touch the pulley?
(ii) If the two tangents from the point where the bucket is attached are equal, state the theorem used.
(iii) Why are the two parts of the rope equal in length?
(iv) If the centre of the pulley is O and the external point is T, prove that OT bisects the angle between the two tangents.

▶ Answer Key Attempt all questions first,
then tap to reveal

Section A

  1. (B) 2. (B) 3. (A) 4. (C) 5. (C)
  2. (C) 7. (A) 8. (A) 9. (B) 10. (B)

Section B

  1. (i) 0 (ii) 1 (iii) 2
  2. (i) one (ii) two
  3. The length of the segment of the tangent from the external point to the point of contact.
  4. All lines through an interior point intersect the circle at two points.
  5. The tangent is perpendicular to the radius (Theorem 10.1).
  6. Tangents from an external point are equal in length.

Section C

  1. Radius = 7 cm (Pythagoras: r = √(25² – 24²))
  2. Chord length = 8 cm (Perpendicular from centre bisects chord; half-chord = 4 cm, height difference = 4 cm → full chord = 8 cm)
  3. Radius = 3 cm (Pythagoras)
  4. ∠PTQ = 70° (Linear pair and isosceles triangle)
  5. Proof using alternate segment or co-interior angles = 180° (marking: statement 1, diagram 1, proof 3)

Section D

  1. Full proof of Theorem 10.2 (RHS congruence) – 5 marks
  2. TP = 20/3 cm (similar triangles or Pythagoras) – 5 marks
  3. AB = 15 cm, AC = 13 cm (tangent segments equal) – 5 marks

Section E

Case 1

(i) Perpendicular (1)
(ii) One (1)
(iii) 90° (1)
(iv) Wheel rolls along tangent line (1)

Case 2

(i) Two (1)
(ii) Theorem 10.2 (1)
(iii) Equal tangents (1)
(iv) OT is angle bisector (1)

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All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.