Vardaan Learning Institute

Chapter Practice Sheet: Circles

Class: 10 (CBSE) Subject: Mathematics Max. Marks: 50
SECTION A: OBJECTIVE TYPE QUESTIONS (1 Mark 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. The radius of the circle is:
  1. 7 cm
  2. 12 cm
  3. 15 cm
  4. 24.5 cm
[1]
2. If TP and TQ are two tangents to a circle with centre O so that $\angle POQ = 110^{\circ}$, then $\angle PTQ$ is equal to:
  1. $60^{\circ}$
  2. $70^{\circ}$
  3. $80^{\circ}$
  4. $90^{\circ}$
[1]
3. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. The radius of the circle is:
  1. 3 cm
  2. 4 cm
  3. 5 cm
  4. 6 cm
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4. The distance between two parallel tangents of a circle of radius 4 cm is:
  1. 2 cm
  2. 4 cm
  3. 6 cm
  4. 8 cm
[1]
5. A point P is 26 cm away from the centre of a circle and the length of the tangent drawn from P to the circle is 24 cm. Find the radius of the circle.
  1. 10 cm
  2. 12 cm
  3. 13 cm
  4. 15 cm
[1]
6. Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
  1. 8 cm
  2. 10 cm
  3. 12 cm
  4. 18 cm
[1]
7. From an external point P, two tangents PA and PB are drawn to a circle with centre O. If $\angle APB = 50^\circ$, then $\angle OAB$ is:
  1. $25^\circ$
  2. $50^\circ$
  3. $100^\circ$
  4. $60^\circ$
[1]
8. At one end A of a diameter AB of a circle of radius 5cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8cm from A is:
  1. 4 cm
  2. 5 cm
  3. 6 cm
  4. 8 cm
[1]
9. In which of the following cases can a circle be inscribed in a quadrilateral ABCD?
  1. $AB+BC = CD+DA$
  2. $AB+CD = BC+DA$
  3. $AB+DA = BC+CD$
  4. $AB-CD = BC-DA$
[1]
10. Assertion (A): If a chord AB subtends an angle of $60^\circ$ at the centre of a circle, then the angle between the tangents at A and B is also $60^\circ$.
Reason (R): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
[1]
SECTION B: SHORT ANSWER TYPE QUESTIONS (2 Marks Each)
11. 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.
[2]
12. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$.
[2]
13. Prove that the parallelogram circumscribing a circle is a rhombus.
[2]
14. In the figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and $\angle BQR = 70^\circ$, then find $\angle AQB$.
[2]
SECTION C: SHORT ANSWER TYPE II QUESTIONS (3 Marks Each)
15. Prove that the lengths of tangents drawn from an external point to a circle are equal.
[3]
16. XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that $\angle AOB = 90^{\circ}$.
[3]
17. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
[3]
18. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle PTQ = 2 \angle OPQ$.
[3]
19. A circle is touching the side BC of $\triangle ABC$ at P and touching AB and AC produced at Q and R respectively. Prove that $AQ = \frac{1}{2} (\text{Perimeter of } \triangle ABC)$.
[3]
20. 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.
[3]
SECTION D: LONG ANSWER TYPE QUESTIONS (5 Marks Each)
21. 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.
[5]
22. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using this, prove that if two tangents are drawn to a circle from an external point, they subtend equal angles at the centre.
[5]
SECTION E: CASE STUDY (4 Marks)
23. Case Study: Playground Design
A circular playground has a radius of 50m. A tangent path is constructed from a point P 120m away from the center O touching the circle at T.

(i) What is the length of the tangent path PT? (2 Marks)
(ii) If another tangent is drawn from P touching the circle at S, what is the length of PS? (1 Mark)
(iii) What is the relation between the radius at the point of contact and the tangent? (1 Mark)
[4]