Class 10 Mathematics
Chapter 10: Circles
50 Practice Questions — All Types Covered
Topics: Tangents | Secants | Theorems | Proofs | Applications
SECTION A — Multiple Choice Questions (1 Mark Each)
1. The number of tangents that can be drawn to a circle from a point outside the circle is:
(a) 0 (b) 1 (c) 2 (d) Infinite
2. A tangent to a circle is perpendicular to the radius at the point of:
(a) Centre (b) Contact (c) Intersection (d) None of these
3. If two tangents are drawn to a circle of radius 5 cm from a point 13 cm away from the centre, the
length of each tangent is:
(a) 8 cm (b) 10 cm (c) 12 cm (d) 6 cm
4. The angle between two tangents drawn from an external point to a circle is 60°. The angle
subtended by the line segment joining the points of tangency at the centre is:
(a) 60° (b) 90° (c) 120° (d) 150°
5. If PA and PB are two tangents from point P to a circle with centre O, then ∠APB + ∠AOB =
(a) 90° (b) 120° (c) 180° (d) 360°
6. A line that intersects a circle in exactly one point is called a:
(a) Secant (b) Chord (c) Tangent (d) Diameter
7. From an external point, two tangents of length (3x + 2) cm and (x + 10) cm are drawn. The value of
x is:
(a) 2 (b) 3 (c) 4 (d) 5
8. The length of a tangent drawn from a point 10 cm away from the centre of a circle of radius 6 cm is:
(a) 4 cm (b) 6 cm (c) 8 cm (d) 16 cm
9. At how many points can a tangent touch a circle?
(a) 1 (b) 2 (c) 3 (d) 0
10. OA is the radius and PA is a tangent to a circle at A. If OA = 5 cm and OP = 13 cm, then PA =
(a) 8 cm (b) 10 cm (c) 12 cm (d) 6 cm
SECTION B — Very Short Answer Questions (2 Marks Each)
11. Define a tangent to a circle. How many tangents can be drawn at a given point on a circle?
12. If two tangents PA and PB are drawn from point P to a circle with centre O, prove that PA = PB.
, 13. A tangent PQ at point P of a circle of radius 5 cm meets a line through the centre O at Q so that
OQ = 12 cm. Find PQ.
14. Prove that the tangent to a circle is perpendicular to the radius through the point of contact.
15. In the figure, if TP and TQ are two tangents to a circle with centre O such that ∠POQ = 110°, find
∠PTQ.
16. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle
which is tangent to the smaller circle.
17. If the angle between two radii of a circle is 130°, find the angle between the tangents at the ends
of those radii.
18. A chord PQ of a circle is parallel to the tangent drawn at point R. Prove that R bisects arc PQ.
19. From a point P, the length of the tangent to a circle is 24 cm and the distance of P from the centre
is 25 cm. Find the radius of the circle.
20. Prove that the lengths of tangents drawn from an external point to a circle are equal.
SECTION C — Short Answer Questions (3 Marks Each)
21. In the figure, two tangents RQ and RP are drawn from an external point R to a circle with centre
O. If ∠PRQ = 120°, prove that OR = PR + RQ.
22. A circle touches all four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.
23. Prove that the parallelogram circumscribing a circle is a rhombus.
24. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that
∠PTQ = 2∠OPQ.
25. A circle of radius 3 cm is drawn inside a triangle with sides 6 cm, 8 cm, and 10 cm. Find the area
of the triangle using the in-radius formula, and verify it.
26. In the figure, AB is a chord of the circle and AOC is its diameter, such that ∠ACB = 50°. If AT is
the tangent to the circle at point A, find ∠BAT.
27. In a right triangle ABC, right-angled at B, BC = 5 cm and AB = 12 cm. The circle is inscribed in the
triangle. Find the radius of the inscribed circle.
28. Prove that in two concentric circles, the chord of the larger circle, which is a tangent to the smaller
circle, is bisected at the point of contact.
29. From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the
tangent to the circle at point E and PA = 14 cm, find the perimeter of triangle PCD.
30. AB is a diameter and AC is a chord of a circle such that ∠BAC = 30°. If the tangent at C intersects
AB extended at D, prove that BC = BD.
SECTION D — Long Answer Questions (4–5 Marks Each)
31. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of
contact. Hence, find the length of tangent drawn from a point 10 cm away from the centre of a circle
of radius 6 cm.
Chapter 10: Circles
50 Practice Questions — All Types Covered
Topics: Tangents | Secants | Theorems | Proofs | Applications
SECTION A — Multiple Choice Questions (1 Mark Each)
1. The number of tangents that can be drawn to a circle from a point outside the circle is:
(a) 0 (b) 1 (c) 2 (d) Infinite
2. A tangent to a circle is perpendicular to the radius at the point of:
(a) Centre (b) Contact (c) Intersection (d) None of these
3. If two tangents are drawn to a circle of radius 5 cm from a point 13 cm away from the centre, the
length of each tangent is:
(a) 8 cm (b) 10 cm (c) 12 cm (d) 6 cm
4. The angle between two tangents drawn from an external point to a circle is 60°. The angle
subtended by the line segment joining the points of tangency at the centre is:
(a) 60° (b) 90° (c) 120° (d) 150°
5. If PA and PB are two tangents from point P to a circle with centre O, then ∠APB + ∠AOB =
(a) 90° (b) 120° (c) 180° (d) 360°
6. A line that intersects a circle in exactly one point is called a:
(a) Secant (b) Chord (c) Tangent (d) Diameter
7. From an external point, two tangents of length (3x + 2) cm and (x + 10) cm are drawn. The value of
x is:
(a) 2 (b) 3 (c) 4 (d) 5
8. The length of a tangent drawn from a point 10 cm away from the centre of a circle of radius 6 cm is:
(a) 4 cm (b) 6 cm (c) 8 cm (d) 16 cm
9. At how many points can a tangent touch a circle?
(a) 1 (b) 2 (c) 3 (d) 0
10. OA is the radius and PA is a tangent to a circle at A. If OA = 5 cm and OP = 13 cm, then PA =
(a) 8 cm (b) 10 cm (c) 12 cm (d) 6 cm
SECTION B — Very Short Answer Questions (2 Marks Each)
11. Define a tangent to a circle. How many tangents can be drawn at a given point on a circle?
12. If two tangents PA and PB are drawn from point P to a circle with centre O, prove that PA = PB.
, 13. A tangent PQ at point P of a circle of radius 5 cm meets a line through the centre O at Q so that
OQ = 12 cm. Find PQ.
14. Prove that the tangent to a circle is perpendicular to the radius through the point of contact.
15. In the figure, if TP and TQ are two tangents to a circle with centre O such that ∠POQ = 110°, find
∠PTQ.
16. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle
which is tangent to the smaller circle.
17. If the angle between two radii of a circle is 130°, find the angle between the tangents at the ends
of those radii.
18. A chord PQ of a circle is parallel to the tangent drawn at point R. Prove that R bisects arc PQ.
19. From a point P, the length of the tangent to a circle is 24 cm and the distance of P from the centre
is 25 cm. Find the radius of the circle.
20. Prove that the lengths of tangents drawn from an external point to a circle are equal.
SECTION C — Short Answer Questions (3 Marks Each)
21. In the figure, two tangents RQ and RP are drawn from an external point R to a circle with centre
O. If ∠PRQ = 120°, prove that OR = PR + RQ.
22. A circle touches all four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.
23. Prove that the parallelogram circumscribing a circle is a rhombus.
24. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that
∠PTQ = 2∠OPQ.
25. A circle of radius 3 cm is drawn inside a triangle with sides 6 cm, 8 cm, and 10 cm. Find the area
of the triangle using the in-radius formula, and verify it.
26. In the figure, AB is a chord of the circle and AOC is its diameter, such that ∠ACB = 50°. If AT is
the tangent to the circle at point A, find ∠BAT.
27. In a right triangle ABC, right-angled at B, BC = 5 cm and AB = 12 cm. The circle is inscribed in the
triangle. Find the radius of the inscribed circle.
28. Prove that in two concentric circles, the chord of the larger circle, which is a tangent to the smaller
circle, is bisected at the point of contact.
29. From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the
tangent to the circle at point E and PA = 14 cm, find the perimeter of triangle PCD.
30. AB is a diameter and AC is a chord of a circle such that ∠BAC = 30°. If the tangent at C intersects
AB extended at D, prove that BC = BD.
SECTION D — Long Answer Questions (4–5 Marks Each)
31. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of
contact. Hence, find the length of tangent drawn from a point 10 cm away from the centre of a circle
of radius 6 cm.
