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Class 10 Maths Chapter 10: Circles — Important Questions & Sample Paper

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Reviewed by qpaper's CBSE curriculum team · Edited by Mohit · Updated June 2026

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Yes — this page has 44+ original Class 10 Mathematics Chapter 10 (“Circles”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Class 10 Mathematics Chapter 10: Circles introduces students to the fundamental properties of tangents to a circle. This chapter builds on earlier concepts of circles and focuses primarily on the relationship between a circle and a line that touches it at exactly one point. Key concepts include: a tangent is perpendicular to the radius at the point of contact; exactly two tangents can be drawn from an external point, and these tangents are equal in length. Students learn to prove and apply these theorems in various geometric configurations. Typical exam questions range from direct application (e.g., finding the length of a tangent given the distance from the centre) to more complex proofs. You might be asked to show that in a quadrilateral circumscribing a circle, the sums of opposite sides are equal. Problems often involve inscribed circles in triangles, where you calculate the segments formed on the sides, or finding angles between tangents based on given central angles. Questions on concentric circles, where a chord of the larger circle is tangent to the smaller, test your understanding of perpendicularity and Pythagoras theorem. Mastering these concepts requires strong geometric reasoning and the ability to identify radii, tangents, and right angles in a diagram. With regular practice, students can confidently tackle the variety of quadrilaterals, triangles, and circle combinations that appear in CBSE board exams.

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Mathematics — Circles

Class 10Time: 3 hrsMax Marks: 80

SECTION A

  1. 1.

    A circle can have at most ________ parallel tangents.

    (a) 0(b) 1(c) 2(d) Infinite
    1
  2. 2.

    A circle is inscribed in a triangle ABC with sides AB = 12 cm, BC = 8 cm, CA = 10 cm. If P, Q, R are points of tangency on BC, CA, AB respectively, then BP is:

    (a) 5 cm(b) 4 cm(c) 3 cm(d) 6 cm
    1
  3. 3.

    A quadrilateral ABCD circumscribes a circle. If AB = 7 cm, BC = 8 cm, CD = 5 cm, then AD is equal to:

    (a) 4 cm(b) 6 cm(c) 7 cm(d) 9 cm
    1

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question typeMarks eachIn our bank
Multiple Choice (MCQ)1 mark13
Assertion–Reason1 mark6
Short Answer2 marks8
Short Answer3 marks6
Long Answer5 marks5
Case Study4 marks6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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  1. Q1. A circle can have at most ________ parallel tangents.

    1 mark
    Multiple Choice (MCQ)
    (A) 0(B) 1(C) 2(D) Infinite
    ▸ Answer

    2

  2. Q2. A circle is inscribed in a triangle ABC with sides AB = 12 cm, BC = 8 cm, CA = 10 cm. If P, Q, R are points of tangency on BC, CA, AB respectively, then BP is:

    1 mark
    Multiple Choice (MCQ)
    (A) 5 cm(B) 4 cm(C) 3 cm(D) 6 cm
    ▸ Answer

    5 cm

  3. Q3. A quadrilateral ABCD circumscribes a circle. If AB = 7 cm, BC = 8 cm, CD = 5 cm, then AD is equal to:

    1 mark
    Multiple Choice (MCQ)
    (A) 4 cm(B) 6 cm(C) 7 cm(D) 9 cm
    ▸ Answer

    4 cm

  4. Q4. Which of the following correctly relates the angle between two tangents drawn from an external point P (∠APB) and the angle subtended by the chord of contact AB at the centre (∠AOB)?

    1 mark
    Multiple Choice (MCQ)
    (A) ∠APB = 2∠AOB(B) ∠APB + ∠AOB = 180°(C) ∠APB = ½∠AOB(D) ∠APB = ∠AOB
    ▸ Answer

    ∠APB + ∠AOB = 180°

  5. Q5. Assertion (A): The angle between the two tangents drawn from an external point to a circle is always acute. Reason (R): The line joining the external point to the centre bisects the angle between the tangents, and the radius is perpendicular to the tangent at the point of contact.

    1 mark
    Assertion–Reason
    (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.
    ▸ Answer

    A is false but R is true.

  6. Q6. State the theorem that relates the tangent at a point of a circle to the radius through the point of contact.

    2 marks
    Short Answer
    ▸ Answer

    The tangent at any point of a circle is perpendicular to the radius through the point of contact.

  7. Q7. PA and PB are tangents from an external point P to a circle with centre O. A third tangent at a point M on the circle (lying on the arc AB nearer to P) intersects PA at C and PB at D. Prove that the perimeter of triangle PCD equals 2PA.

    2 marks
    Short Answer
    ▸ Answer

    Perimeter of triangle PCD = 2PA

  8. Q8. In triangle ABC, right-angled at B, with AB = 24 cm and BC = 10 cm, a circle is inscribed touching all three sides. Find the radius of the incircle.

    3 marks
    Short Answer
    ▸ Answer

    4 cm

  9. Q9. From an external point P, two tangents PA and PB are drawn to a circle with centre O. If ∠AOB = 110°, find ∠APB.

    3 marks
    Short Answer
    ▸ Answer

    70°

  10. Q10. PA and PB are tangents drawn from an external point P to a circle with centre O. Prove that OP is the perpendicular bisector of the chord AB. Hence, or otherwise, if PA = 10 cm and the radius of the circle is 6 cm, find the radius of the circumcircle of triangle PAB.

    5 marks
    Long Answer
    ▸ Answer

    OP is the perpendicular bisector of AB. Radius of circumcircle of triangle PAB = √34 cm.

  11. Q11. Prove that the sum of lengths of a pair of opposite sides of a quadrilateral circumscribing a circle is equal to the sum of lengths of the other pair.

    5 marks
    Long Answer
    ▸ Answer

    AB + CD = AD + BC.

  12. Q12. A boy stands at point P, 10 m away from the centre O of a circular pond of radius 6 m. He holds two rods PA and PB that just touch the surface of the pond at points A and B, forming tangents.

    4 marks
    Case Study
    1. (i) Find the length of one rod, PA.2 marks
    2. (ii) What is the angle between the radius OA and the tangent PA?2 marks
    ▸ Answer

    PA = 8 m; angle OAP = 90°.

Frequently asked questions

How many tangents can be drawn from an external point to a circle?

From an external point, exactly two tangents can be drawn to a circle. These two tangents are equal in length, and the line joining the external point to the centre bisects the angle between the tangents.

What is the key theorem about a quadrilateral circumscribing a circle?

If a quadrilateral circumscribes a circle, the sums of the lengths of opposite sides are equal. That is, AB + CD = BC + AD. This is often proved using the property that tangents drawn from an external point (the vertices of the quadrilateral) to the circle are equal.

How do you find the length of a tangent from an external point when given the distance from the centre and the radius?

Use the Pythagoras theorem. Since the radius to the point of contact is perpendicular to the tangent, you get a right-angled triangle with the radius, tangent, and distance from centre to external point. So, tangent length = √(OP² - r²).

What is the angle between a tangent and the radius at the point of contact?

A tangent to a circle is always perpendicular to the radius drawn to the point of contact. This is a fundamental property used in many geometric proofs and calculations.

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