Chapter 10: Circles

Important questions — the ones the board asked most (20)

1. Q1Short answer3 marksAsked: 2024, 2025, 2026

   Prove that a parallelogram circumscribing a circle, is a rhombus.

   

   Medha Ma’am

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   Model answer · tangents, circumscribed quadrilateral proof

   1. 1.Given: ABCD is a parallelogram circumscribing a circle with centre O. The sides AB, BC, CD and DA touch the circle at the points of contact P, Q, R and S respectively. To prove: ABCD is a rhombus. (Figure: the circle with centre O drawn inside parallelogram ABCD, touching AB at P, BC at Q, CD at R and DA at S.)0.5 marks
   2. 2.The lengths of tangents drawn from an external point to a circle are equal. So, AP = AS (tangents from A) BP = BQ (tangents from B) CR = CQ (tangents from C) DR = DS (tangents from D)1 mark
   3. 3.Adding these equations side by side: AP + BP + CR + DR = AS + BQ + CQ + DS (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ) AB + CD = AD + BC In a parallelogram, opposite sides are equal, so AB = CD and AD = BC. So, 2AB = 2BC1 mark
   4. 4.Therefore, AB = BC. Since AB = BC = CD = AD, all four sides of the parallelogram are equal. Hence ABCD is a rhombus.0.5 marks

   Diagram: Draw a circle with centre O inside parallelogram ABCD so that every side touches the circle. Label the points of contact P on AB, Q on BC, R on CD and S on DA.

   • Give the reason "The lengths of tangents drawn from an external point to a circle are equal" before you write the four equal pairs AP = AS, BP = BQ, CR = CQ, DR = DS.
   • Write "AB = CD and AD = BC" as the property of a parallelogram, then go from 2AB = 2BC to AB = BC.
   • Label the points of contact P, Q, R, S on the figure in the same order you use in the proof.

   Marks split as in the board’s published marking scheme

2. Q2Very short answer2 marksAsked: 2023, 2025, 2026

   Two concentric circles are of radii 5 cm and 4 cm. Find the length of the chord of the larger circle which touches the smaller circle.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 75 s

   Model answer · tangent, concentric circles chord

   1. 1.Given: two concentric circles with centre O, where the larger circle has radius OA = 5 cm and the smaller circle has radius OM = 4 cm. AB is a chord of the larger circle that touches the smaller circle at the point M (figure drawn alongside).0.5 marks
   2. 2.AB is a tangent to the smaller circle at M, and OM is its radius. The tangent to a circle is perpendicular to the radius through the point of contact, so OM ⊥ AB. In right triangle OMA, by the Pythagoras Theorem: AM = √(OA² − OM²) = √(5² − 4²) = √(25 − 16) = √9 = 3 cm1 mark
   3. 3.The perpendicular from the centre bisects the chord, so AM = MB. AB = 2 × AM = 2 × 3 = 6 cm So the length of the chord is 6 cm.0.5 marks

   Diagram: Draw two concentric circles with centre O. Draw chord AB of the larger circle so that it touches the smaller circle at M. Join OM and OA, mark the right angle at M, and write OA = 5 cm and OM = 4 cm.

   • Write the reason for "OM ⊥ AB": the tangent to a circle is perpendicular to the radius through the point of contact.
   • Give the reason for doubling AM: the perpendicular from the centre bisects the chord.
   • On the figure, label O, A, B and M, mark the right angle at M, and write both radii.

   Marks split as in the board’s published marking scheme

3. Q3Very short answer2 marksAsked: 2023, 2024, 2025

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

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 75 s

   Model answer · tangents at ends of diameter

   1. 1.Given: a circle with centre O and diameter AB. Tangent l is drawn at A and tangent m at B. To prove: l || m. (Figure: the circle with centre O, diameter AB, tangent l at A and tangent m at B. Angle 1 is marked between l and AB at A, and angle 2 between m and AB at B, on opposite sides of AB.)0.5 marks
   2. 2.Proof: Tangents l and m are drawn at the end points A and B of the diameter AB of the circle. The tangent at any point of a circle is perpendicular to the radius through the point of contact. So OA ⊥ l, which gives ∠1 = 90°, and OB ⊥ m, which gives ∠2 = 90°.0.5 marks
   3. 3.Therefore, ∠1 = ∠2 (each is 90°).0.5 marks
   4. 4.But these are alternate interior angles made by the transversal AB with the lines l and m. Hence l || m.0.5 marks

   Diagram: Draw a circle with centre O and diameter AB. Draw tangent l at A and tangent m at B. Mark ∠1 between l and AB at A, and ∠2 between m and AB at B, on opposite sides of AB.

   • Mark ∠1 and ∠2 on opposite sides of AB in the figure, so that they are alternate angles.
   • Give the reason for ∠1 = 90° and ∠2 = 90°: the tangent is perpendicular to the radius through the point of contact.
   • End with "these are alternate interior angles", then write l || m.

   Marks split as in the board’s published marking scheme

4. Q4Short answer3 marksAsked: 2025, 2026

   Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

   How the model answer begins

   1. Figure: A circle with centre O and a quadrilateral PQRS drawn around it. The sides PQ, QR, RS and SP touch the circle at A, B, C and D. O is joined to A, B, C, D, P, Q, R and S. The angles at O are marked ∠1 = ∠DOS, ∠2…

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5. Q5Long answer5 marksAsked: 2025

   (a) The given figure shows a circle with centre O and radius 4 cm circumscribed by triangle ABC. BC touches the circle at D such that BD = 6 cm, DC = 10 cm. Find the length of AE. OR (b) PA and PB are tangents drawn to a circle with centre O. If angle AOB = 120 degree and OA = 10 cm, then (i) Find angle OPA. (ii) Find the perimeter of triangle OAP. (iii) Find the length of chord AB.

   How the model answer begins

   1. Given: a circle with centre O and radius 4 cm touches BC at D, CA at E and AB at F. BD = 6 cm and DC = 10 cm. Let AE = x cm. Then AF = x cm, CE = CD = 10 cm and BF = BD = 6 cm. So BC = 16 cm, CA = (10 + x) cm and AB =…

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6. Q6Long answer5 marksAsked: 2025

   (a) The given figure shows a circle with centre O and radius 4 cm circumscribed by ΔABC. BC touches the circle at D such that BD = 6 cm, DC = 10 cm. Find the length of AE. OR (b) PA and PB are tangents drawn to a circle with centre O. If ∠AOB = 120° and OA = 10 cm, then (i) Find ∠OPA. (ii) Find the perimeter of ΔOAP. (iii) Find the length of chord AB.

   How the model answer begins

   1. (b) Given: PA and PB are tangents from the external point P to a circle with centre O, ∠AOB = 120° and OA = 10 cm. (i) The tangent to a circle is perpendicular to the radius through the point of contact. So ∠OAP = ∠OBP…

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7. Q7Long answer5 marksAsked: 2024

   (a) In the given figure, AB is chord of length 6 cm of a circle of radius 5 cm. The tangents at A and B intersect at a point P. Find the length of PB. OR (b) Prove that the parallelogram circumscribing a circle is a rhombus. Also, find area of the rhombus, if radius of circle is 3 cm and length of one side of the rhombus is 10 cm.

   How the model answer begins

   1. Given: O is the centre of the circle, the radius OA = OB = 5 cm, the chord AB = 6 cm, and the tangents at A and B meet at the external point P. Join OP and let it cut AB at Q. The lengths of tangents drawn from an…

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8. Q8Long answer5 marksAsked: 2023

   (a) A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the sides AB and AC, if it is given that the area of triangle ABC is 90 cm^2. or (b) There are two circles with centres O and O' having radii 6 cm and 8 cm respectively. They intersect each other at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.

   How the model answer begins

   1. (a) Let the circle touch BC, CA, AB at D, E, F. Tangents from a point are equal: AF = AE = x, BF = BD = 10 cm, CE = CD = 8 cm. OR (b) OP and O′P are tangents to the other circles, so OP ⊥ O′P: ∠OPO′ = 90°.

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9. Q9Long answer5 marksAsked: 2026

   PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

   How the model answer begins

   1. Given: a circle with centre O and radius 5 cm. PQ and PR are tangents from the external point P that touch the circle at Q and R, and OP = 13 cm, OQ = 5 cm. The tangent to a circle is perpendicular to the radius through…

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10. Q10Long answer5 marksAsked: 2023

    34. (a) A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if it is given that area triangle ABC = 90 cm^2. OR (b) Two circles with centres O and O' of radii 6 cm and 8 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.

    How the model answer begins

    1. (a) Tangents from a point are equal: AF = AE = x, BF = BD = 10 cm, CE = CD = 8 cm. OR (b) Radius ⊥ tangent, so OP ⊥ O′P and ∠OPO′ = 90°.

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11. Q11Long answer5 marksAsked: 2023

    (A) Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ. OR (B) A circle touches the side BC of a triangle ABC at a point P and touches AB and AC when produced at Q and R respectively. Show that AQ = 1/2 (Perimeter of triangle ABC).

    How the model answer begins

    1. (A) Given: tangents TP and TQ from an external point T to a circle with centre O. To prove: ∠PTQ = 2∠OPQ. OR (B) Given: a circle touches BC at P and AB, AC produced at Q, R. To show: AQ = ½ (perimeter of ΔABC).

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12. Q12Long answer5 marksAsked: 2023

    33. (A) Prove that a parallelogram circumscribing a circle is a rhombus. OR (B) In the given figure, tangents PQ and PR are drawn to a circle such that angle RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the measure of angle RQS.

    How the model answer begins

    1. Given: ABCD is a parallelogram circumscribing a circle. The sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. To prove: ABCD is a rhombus. (Figure: a circle with the parallelogram ABCD…

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13. Q13Case-based4 marksAsked: 2024

    A backyard is in the shape of a triangle ABC with right angle at B. AB = 7 m and BC = 15 m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that AP = x m. Based on the above information, answer the following questions : (i) Find the length of AR in terms of x. (ii) Write the type of quadrilateral BQOR. (iii) (a) Find the length PC in terms of x and hence find the value of x. OR (b) Find x and hence find the radius r of circle.

    How the model answer begins

    1. (i) AR = x m. A is an external point to the circle, and AP and AR are the two tangents from A, with P and R as the points of contact. The lengths of the two tangents from an external point to a circle are equal, so AR =…

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14. Q14Case-based4 marksAsked: 2023

    The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around one and a half times through a circle, then releases the throw. When released, the discus travels along tangent to the circular spin orbit. In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and angle ABO = 30 degree. PQ is parallel to OA. Based on above information : (a) find the length of AB. (b) find the length of OB. (c) find the length of AP. OR find the length of PQ.

15. Q15Case-based4 marksAsked: 2023

    In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and angle ABO = 30 degree. PQ is parallel to OA. Based on above information : (a) find the length of AB. (b) find the length of OB. (c) find the length of AP. OR find the length of PQ.

    How the model answer begins

    1. (a) OA ⊥ AB (radius ⊥ tangent), so in ΔOAB, tan 30° = OA/AB = 75/AB, giving AB = 75√3 cm.

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16. Q16Case-based4 marksAsked: 2023

    Case Study - 2 37. Circles play an important part in our life. When a circular object is hung on the wall with a cord at nail N, the cords NA and NB work like tangents. Observe the figure, given that ∠ ANO = 30° and OA = 5 cm. Based on the above, answer the following questions : (i) Find the distance AN. (ii) Find the measure of ∠ AOB. (iii) (a) Find the total length of cords NA, NB and the chord AB. OR (iii) (b) If ∠ ANO is 45°, then name the type of quadrilateral OANB. Justify your answer.

    How the model answer begins

    1. In right triangle OAN, since OA is a radius and NA is a tangent, OA ⊥ NA by the theorem that the tangent to a circle is perpendicular to the radius through the point of contact. Therefore, angle OAN = 90°. Using tan 30°…

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17. Q17Case-based4 marksAsked: 2023

    Circles play an important part in our life. When a circular object is hung on the wall with a cord at nail N, the cords NA and NB work like tangents. Observe the figure, given that < ANO = 30° and OA = 5 cm. Based on the above, answer the following questions : (i) Find the distance AN. 1 (ii) Find the measure of < AOB. 1 (iii) (a) Find the total length of cords NA, NB and the chord AB. 2 OR (iii) (b) If < ANO is 45°, then name the type of quadrilateral OANB. Justify your answer. 2

    How the model answer begins

    1. (i) Radius ⊥ tangent, so ∠OAN = 90°; tan 30° = OA/AN gives 1/√3 = 5/AN.

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18. Q18Case-based4 marksAsked: 2023

    Case Study - 1 36. Circles play an important role in our lives. When a circular object is hung on a wall with a string on a nail N, the strings NA and NB act like tangents. Observe the figure, given that angle ANO = 30° and OA = 5 cm. Based on the above, answer the following questions : (i) Find the length of AN. (ii) Find the measure of angle AOB. (iii) (a) Find the total length of the strings NA, NB and the chord AB. or (iii) (b) If angle ANO = 45°, write the name of the type of quadrilateral OANB. Justify your answer.

    How the model answer begins

    1. (i) Given: O is the centre of the circle, NA and NB are tangents from the external point N, ∠ANO = 30° and OA = 5 cm. The tangent to a circle is perpendicular to the radius through the point of contact, so ∠OAN = 90°.…

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19. Q19Case-based4 marksAsked: 2024

    The picture given below shows a circular mirror hanging on the wall with a cord. The diagram represents the mirror as a circle with centre O. AP and AQ are tangents to the circle at P and Q respectively such that AP = 30 cm and angle PAQ = 60°. Based on the above information, answer the following questions : (i) Find the length PQ. (ii) Find m angle POQ. (iii) (a) Find the length OA. OR (b) Find the radius of the mirror.

    How the model answer begins

    1. (i) Given: AP and AQ are tangents from the external point A, with AP = 30 cm and ∠PAQ = 60°. The lengths of tangents drawn from an external point to a circle are equal, so AP = AQ. This makes triangle APQ isosceles, so…

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20. Q20Case-based4 marksAsked: 2026

    In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle. Based on above information, answer the following questions : (i) Find m angle AOP. (1) (ii) Prove that triangle OAP is congruent to triangle OBP. (1) (iii) (a) Find the length of fencing required to protect the statue. (Take sqrt(3) = 1.73) (2) OR (b) Find area of quadrilateral OAPB. (Take sqrt(3) = 1.73)

    How the model answer begins

    1. (i) AP touches the inner circle at A, so OA ⊥ AP; in ΔOAP, cos ∠AOP = OA/OP = 7/14 = ½, so ∠AOP = 60°.

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63 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   If two tangents TP and TQ are drawn to a circle with centre O from an external point T such that angle POQ = 120 degrees, then angle PTQ is equal to :

   • (A) 60 degrees ✓
   • (B) 70 degrees
   • (C) 80 degrees
   • (D) 90 degrees

   Answer: (A)

2. 2. CBSE 2026

   In the given figure, PA is a tangent from an external point P to a circle with centre O. If angle POB = 125 degrees, then angle APO is equal to :

   • (A) 25 degrees
   • (B) 65 degrees
   • (C) 90 degrees
   • (D) 35 degrees ✓

   Answer: (D)

3. 3. CBSE 2026

   If TP and TQ are two tangents to a circle with centre O from an external point T so that angle POQ = 120 degree, then angle PTQ is equal to :

   • (A) 60 degree ✓
   • (B) 70 degree
   • (C) 80 degree
   • (D) 90 degree

   Answer: (A)

4. 4. CBSE 2026

   In the given figure, PA and PB are tangents to a circle centred at O. If angle OAB = 15 degree, then angle APB equals :

   • (A) 30 degree ✓
   • (B) 15 degree
   • (C) 45 degree
   • (D) 10 degree

   Answer: (A)

5. 5. CBSE 2026

   In the given figure, PA and PB are tangents to a circle centred at O. If angle AOB = 130 degree, then angle APB equals :

   • (A) 130 degree
   • (B) 50 degree ✓
   • (C) 120 degree
   • (D) 90 degree

   Answer: (B)

6. 6. CBSE 2026

   In the given figure, PA and PB are tangents to a circle centred at O. If ∠ AOB = 130°, then ∠ APB is equal to :

   • (A) 130°
   • (B) 50° ✓
   • (C) 120°
   • (D) 90°

   Answer: (B)

7. 7. CBSE 2026

   In the given figure, PA and PB are tangents to a circle centred at O. If angle AOB = 130°, then angle APB is equal to :

   • (A) 130°
   • (B) 50° ✓
   • (C) 120°
   • (D) 90°

   Answer: (B)

8. 8. CBSE 2026

   In the given figure, two tangents PQ and PR are drawn to a circle with centre O and radius 3 cm. If angle QPR = 60 degree, then the length of each tangent is :

   • (A) 3 sqrt(3) cm ✓
   • (B) 3 cm
   • (C) 6 cm
   • (D) sqrt(3) cm

   Answer: (A)

9. 9. CBSE 2026

   Assertion (A) : Radius is the smallest distance of a tangent from the centre of the circle. Reason (R) : Radius is perpendicular to the tangent.

   • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). ✓
   • (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
   • (C) Assertion (A) is true, but Reason (R) is false.
   • (D) Assertion (A) is false, but Reason (R) is true.

   Answer: (A)

10. 10. CBSE 2026

    In the given figure, PT is a tangent to the circle with centre O and radius r. If angle POT = 45°, then the length of OP is :

    • (A) r*sqrt(2) ✓
    • (B) sqrt(2)r
    • (C) 2r
    • (D) r^2

    Answer: (A)

Questions parents and students ask

How many marks does "Circles" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 7 marks and about 4 questions a year; by marks it ranks 2 in Maths.

Which are the most important (VVI) questions of "Circles"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 83 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.