CBSE Class 10 Circles Sure Shot Questions Set A

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1. Prove that “The tangent at any point of a circle is perpendicular to the radius through the point of contact”.

2. Prove that “The lengths of tangents drawn from an external point to a circle are equal.”

3. Prove that “The centre lies on the bisector of the angle between the two tangents drawn from an external point to a circle.”

4. Find the length of the tangent drawn to a circle of radius 3 cm, from a point distant 5 cm from the centre.

5. A point P is at a distance 13 cm from the centre C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.

6. From appoint Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre of the circle is 25 cm. Find the radius of the circle.

7. The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.

8. Prove that in two concentric circles, the chord of the bigger circle, which touches the smaller circle is bisected at the point of contact.

9. ΔPQR circumscribes a circle of radius r such that angle Q = 90⁰ , PQ = 3 cm and QR = 4 cm. Find r.

10. Prove that the parallelogram circumscribing a circle is a rhombus.

OR

If all the sides of a parallelogram touch the circle, show that the parallelogram is a rhombus.

11. ABC is an isosceles triangle in which AB = AC, circumscribed about a circle. Show that BC is bisected at the point of contact.

12. In Fig., a circle is inscribed in a quadrilateral ABCD in which <B=90⁰ . If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius (r) of the circle. 

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. 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.

26. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

27. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

28. 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.

29. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC

30. Prove that the angle between the 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.

31. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
OR
A circle touches all the four sides a quadrilateral ABCD. Prove that the angles subtended at the centre of the circle by the opposite sides are supplementary.

32. PA and PB are the two tangents to a circle with centre O in which OP is equal to the diameter of the circle. Prove that APB is an equilateral triangle.

33. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the center of the circle.

34. If PQ and RS are two parallel tangents to a circle with centre O and another tangent X, with point of contact C intersects PQ at A and RS at B. Prove that ∠AOB = 90o.

35. The incircle of ΔABC touches the sides BC, CA and AB at D, E and F respectively. If AB = AC, prove that BD = DC.

36. Two tangents PA and PB are drawn to the circle with center O, such that ΔAPB = 1200. Prove that OP = 2AP.

37. A circle is touching the side BC of ΔABC at P and is touching AB and AC when produced at Q and R respectively. Prove that AQ = ½ (Perimeter of Δ ABC).

38. 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.

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