Circles: Class 10 Important Questions with Answers

10.4 Angle MCQs from a figure

CBSE 2023 · 30/2/1 In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If ∠BAC = 65°, then find the measure of ∠BOC. [2]

∠BOC = 180° − 65° = 115° (∠OBA = ∠OCA = 90° in quadrilateral ABOC).

CBSE 2025 · 30/1/2 NCERT 10.2 Q3 If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other at an angle of 80°, then the measure of ∠POA is: (A) 40° (B) 50° (C) 60° (D) 80°

∠APO = 40°, ∠OAP = 90° ⇒ ∠POA = 50° ⇒ (B)

CBSE 2024 · 30/5/1 In the adjoining figure, PA and PB are tangents to a circle with centre O. The reflex angle AOB is 210°. The measure of angle APB is: (A) 210° (B) 150° (C) 105° (D) 30°

∠AOB = 360° − 210° = 150° ⇒ ∠APB = 30° ⇒ (D)

CBSE 2024 · 30/4/1 In the given figure, AB and AC are tangents to the circle. If ∠ABC = 42°, then the measure of ∠BAC is: (A) 96° (B) 42° (C) 106° (D) 86°

AB = AC ⇒ ∠ACB = 42° ⇒ ∠BAC = 180° − 84° = 96° (A)

CBSE 2024 · 30/5/1 In the given figure, RJ and RL are two tangents to the circle. If ∠RJL = 42°, then the measure of ∠JOL is: (A) 42° (B) 84° (C) 96° (D) 138°

RJ = RL ⇒ ∠RLJ = 42°, ∠JRL = 96° ⇒ ∠JOL = 180° − 96° = 84° (B)

CBSE 2025 · 30/1/1 2025 · 30/1/2 In the given figure, PA is a tangent from an external point P to a circle with centre O. If ∠POB = 115°, then ∠APO is equal to: (A) 25° (B) 65° (C) 90° (D) 35°

AOB is a diameter: ∠AOP = 180° − 115° = 65°; in right △OAP ∠APO = 25° (A)

CBSE 2025 · 30/3/1 In the given figure, if PT is a tangent to a circle with centre O and ∠TPO = 35°, then the measure of ∠x (∠TOQ, Q the far end of the diameter through T) is: (A) 110° (B) 115° (C) 120° (D) 125°

∠TOP = 90° − 35° = 55° ⇒ x = 180° − 55° = 125° (D)

CBSE 2025 · 30/3/1 O is the centre; MN is a chord and the tangent ML at M makes 70° with MN. ∠MON is: (A) 120° (B) 140° (C) 70° (D) 90°

∠OMN = 90° − 70° = 20° = ∠ONM ⇒ ∠MON = 140° (B)

CBSE 2025 · 30/2/1 RS is the tangent at L and MN is the diameter. If ∠NML = 30°, then ∠RLM is: (A) 30° (B) 60° (C) 90° (D) 120°

OM = OL ⇒ ∠OLM = 30°; ∠OLR = 90° ⇒ ∠RLM = 90° − 30° = 60° (B)

HOTS (higher-order) practice

H1. Two tangents PA, PB from P to a circle of radius 5 cm make 60° with each other. Find OP and the area of quadrilateral OAPB.

H2. A circle touches all four sides of a quadrilateral with AB = 6, BC = 7, CD = 4 cm. Find AD.

Case-based practice

20 MCQs (1 mark each)

1. The number of tangents from a point inside a circle is
   (a) 0 (b) 1 (c) 2 (d) infinite
2. The angle between a tangent and the radius at the point of contact is
   (a) 60° (b) 90° (c) 45° (d) 180°
3. A tangent from a point 13 cm from the centre of a circle of radius 5 cm has length
   (a) 12 cm (b) 8 cm (c) 18 cm (d) √194 cm
4. If ∠APB = 70° for tangents PA, PB, then ∠AOB is
   (a) 70° (b) 110° (c) 140° (d) 20°
5. A circle can have at most ___ parallel tangents.
   (a) 1 (b) 2 (c) 3 (d) infinitely many
6. PA = 6 cm is a tangent; PB is the other tangent. PB is
   (a) 3 cm (b) 6 cm (c) 12 cm (d) cannot say
7. If tangents PA, PB make ∠APB = 90° and r = 4 cm, then PA is
   (a) 4 cm (b) $4\sqrt{2}$ cm (c) 8 cm (d) 2 cm
8. A quadrilateral ABCD circumscribes a circle with AB = 5, BC = 6, CD = 7. AD is
   (a) 6 (b) 8 (c) 4 (d) 5
9. In the figure, ∠AOB = 130° (A, B points of contact). ∠APB is
   (a) 50° (b) 130° (c) 65° (d) 25°
10. Two concentric circles have radii 13 cm and 5 cm. The chord of the larger touching the smaller has length
    (a) 12 cm (b) 24 cm (c) 18 cm (d) 8 cm
11. If ∠APO = 25° (PA tangent, O centre), ∠AOP is
    (a) 25° (b) 65° (c) 90° (d) 155°
12. The common tangents of two circles touching externally number
    (a) 1 (b) 2 (c) 3 (d) 4
13. A line meeting a circle at two points is a
    (a) tangent (b) chord (c) secant (d) radius
14. OP = 2r for a point P and circle radius r. ∠APB between the tangents is
    (a) 30° (b) 60° (c) 90° (d) 120°
15. A parallelogram circumscribing a circle is a
    (a) rectangle (b) rhombus (c) trapezium (d) kite only
16. If PA, PB are tangents and ∠PAB = 50°, then ∠APB is
    (a) 50° (b) 80° (c) 100° (d) 130°
17. A triangle circumscribes a circle of radius 2 cm and has perimeter 30 cm. Its area is
    (a) 60 cm² (b) 30 cm² (c) 15 cm² (d) 120 cm²
18. The length of a tangent from P is 15 cm and OP = 17 cm. The diameter is
    (a) 8 cm (b) 16 cm (c) 32 cm (d) 4 cm
19. The tangents at the ends of a diameter are
    (a) perpendicular (b) parallel (c) intersecting (d) equal to the diameter
20. If the radius is 3 cm and tangents from P are at 60°, OP is
    (a) 6 cm (b) 3√3 cm (c) 3 cm (d) 9 cm

Answer key

1 (a) · 2 (b) · 3 (a) · 4 (b) · 5 (b) · 6 (b) · 7 (a) · 8 (a) · 9 (a) · 10 (b) · 11 (b) · 12 (c) · 13 (c) · 14 (b) · 15 (b) · 16 (b) · 17 (b) · 18 (b) · 19 (b) · 20 (a)

Hints: Q7 OAPB is a square. Q8 5 + 7 = AD + 6. Q14 $\sin \angle APO=\frac{1}{2}$ ⇒ 30°, doubled 60°. Q16 ∠PBA = 50°. Q17 $r\times s=2\times 15$. Q18 r = 8. Q20 $\frac{3}{\sin 30°}$.

Assertion–Reason (10)

(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.

1. A: PA = PB for tangents from P. R: △OAP ≅ △OBP by RHS.
2. A: From a point inside a circle, two tangents can be drawn. R: A tangent meets the circle at exactly one point.
3. A: If ∠APB = 60°, then ∠AOB = 120°. R: ∠APB + ∠AOB = 180°.
4. A: Tangents at the ends of a diameter are parallel. R: Both are perpendicular to the same line.
5. A: A rectangle can circumscribe a circle only if it is a square. R: For a circumscribed quadrilateral AB + CD = AD + BC.
6. A: The chord of the larger of two concentric circles (radii 10, 6) touching the smaller is 16 cm. R: The radius to the point of contact bisects that chord.
7. A: A circle has exactly two tangents. R: Through each point of a circle there is exactly one tangent.
8. A: If PA = 12 cm is a tangent from P and r = 5 cm, then OP = 13 cm. R: $O{P}^2=O{A}^2+P{A}^2$.
9. A: The angle between the tangent and the chord through the point of contact is always 90°. R: Tangent ⊥ radius.
10. A: Two circles intersecting at two points have two common tangents. R: Two circles can have at most four common tangents.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (a) · 5 (a) · 6 (a) · 7 (d) · 8 (a) · 9 (d) · 10 (b)

Q5: AB + CD = AD + BC with AB = CD, AD = BC ⇒ AB = BC ⇒ square. Q9: only a chord that is a diameter makes 90° with the tangent.

Very short answer (1 mark each)

1. How many tangents can be drawn from a point on the circle? Ans. 1
2. A tangent from P is 8 cm; OP = 10 cm. Find r. Ans. 6 cm
3. If ∠AOB = 100° (A, B contacts), find ∠APB. Ans. 80°
4. Name the theorem proved using RHS congruence. Ans. Tangents from an external point are equal.
5. Quadrilateral circumscribing a circle: AB = 8, CD = 6, BC = 9. Find AD. Ans. 5
6. Tangents PA, PB with ∠APB = 90°. What figure is OAPB? Ans. a square
7. Two circles touch internally. How many common tangents? Ans. 1
8. Write the area of a triangle circumscribing a circle of radius r with semi-perimeter s. Ans. r × s

Short answer practice (2 marks each)

1. From a point P, 13 cm from the centre O of a circle of radius 5 cm, tangents PA and PB are drawn. Find the area of quadrilateral OAPB.

2. Prove that the centre of a circle touching two intersecting lines lies on the bisector of the angle between them.

Short answer practice (3 marks each)

3. A circle is inscribed in △ABC with AB = 10, BC = 8, CA = 12 cm, touching AB, BC, CA at D, E, F. Find AD, BE and CF.

4. Two tangents TP and TQ are drawn from T to a circle with centre O, and ∠PTQ = 50°. Find ∠OPQ and ∠POQ.

5. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Long answer (5 marks)

6. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using it, if a circle touches the side BC of △ABC at P and AB, AC produced at Q and R, prove that AQ = ½(perimeter of △ABC).