CBSE Class 10 Circles Sure Shot Questions Set 01

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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.

Question. In the above question (no. 3) what is the ratio of AM : BM?
(a) 5 : 6
(b) 3 : 2
(c) 1 : 1
(d) can't be determined
Answer: (c) 1 : 1

Question. If the diagonals of a cyclic quadrilateral are equal, then the quadrilateral is
(a) rhombus
(b) square
(c) rectangle
(d) none of the options
Answer: (c) rectangle

Question. The quadrilateral formed by angle bisectors of a cyclic quadrilateral is a:
(a) rectangle
(b) square
(c) parallelogram
(d) cyclic quadrilateral
Answer: (d) cyclic quadrilateral

Question. Three circles touch each other externally. The distance between their centre is 5 cm, 6 cm and 7 cm. Find the radii of the circles :
(a) 2 cm, 3 cm, 4 cm
(b) 3 cm, 4 cm, 1 cm
(c) 1 cm, 2.5 cm, 3.5 cm
(d) 1 cm, 2 cm, 4 cm
Answer: (a) 2 cm, 3 cm, 4 cm

Question. O and O' are the centres of two circles which touch each other externally at P. AB is a common tangent. Find \( \angle APO \):
(a) 90°
(b) 120°
(c) 60°
(d) data insufficient
Answer: (a) 90°

Question. If AB is a chord of a circle, P and Q are two points on the circle different from A and B, then:
(a) the angle subtended by AB at P and Q are either equal or supplementary.
(b) the sum of the angles subtended by AB at P and Q is always equal two right angles.
(c) the angles subtended at P and Q by AB are always equal.
(d) the sum of the angles subtended at P and Q is equal to four right angles.
Answer: (a) the angle subtended by AB at P and Q are either equal or supplementary.

Question. In a circle of radius 5 cm, AB and AC are the two chords such that AB = AC = 6 cm. Find the length of the chord BC.
(a) 4.8 cm
(b) 10.8 cm
(c) 9.6 cm
(d) none of the options
Answer: (c) 9.6 cm

Question. In a circle of radius 17 cm, two parallel chords are drawn on opposite sides of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is :
(a) 23 cm
(b) 30 cm
(c) 15 cm
(d) none of the options
Answer: (b) 30 cm

Question. A circle has two parallel chords of lengths 6 cm and 8 cm. If the chords are 1 cm apart and the centre is on the same side of the chords, then a diameter of the circle is of length:
(a) 5 cm
(b) 6 cm
(c) 8 cm
(d) 10 cm
Answer: (d) 10 cm

Question. Three equal circles of unit radius touch each other. Then, the area of the circle circumscribing the three circles is :
(a) \( 6\pi (2 + \sqrt{3})^2 \)
(b) \( \frac{\pi}{6}(2 + \sqrt{3})^2 \)
(c) \( \frac{\pi}{3}(2 + \sqrt{3})^2 \)
(d) \( 3\pi (2 + \sqrt{3})^2 \)
Answer: (c) \( \frac{\pi}{3}(2 + \sqrt{3})^2 \)

Question. Through any given set of four points P,Q, R, S it is possible to draw :
(a) atmost one circle
(b) exactly one circle
(c) exactly two circles
(d) exactly three circles
Answer: (a) atmost one circle

Question. The number of common tangents that can be drawn to two given circles is at the most :
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. In the diagram, PQ and QR are tangents to the circle centre O, at P and R respectively. Find the value of x.
(a) 25
(b) 35
(c) 45
(d) 55
Answer: (d) 55

Question. AB and CD are two parallel of a circle such that AB=10 cm and CD = 24 cm. The chords are on opposite sides of the centre and the distance between them is 17 cm. Find the radius of the circle
(a) 11 cm
(b) 12 cm
(c) 13 cm
(d) 14 cm
Answer: (c) 13 cm

Question. An equilateral triangle has side \( 2\sqrt{3} \) cm. The radius of its circumcircle will be
(a) 2 cm
(b) \( \sqrt{3} \) cm
(c) 3 cm
(d) 4 cm
Answer: (a) 2 cm

Question. Find the distance of a perpendicular from the centre of a circle to the chord if the diameter of the circle is 30 cm and its chord is 24 cm.
(a) 6 cm
(b) 7 cm
(c) 9 cm
(d) 10 cm
Answer: (c) 9 cm

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