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CBSE Notes Class 10 2020 : Mathematics (Vidhyalakshmi School, North Arcot) Circles

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VIDHYALAKSHMI SENIOR SECONDARY SCHOOL Circles MATHEMATICS Class : X Roll No. : Date : 9.10.19 Time MM - 159 1. Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25 cm. Given that radius of the circle is 7 cm. 1 2. What is the angle between a tangent to a circle and the radius through the point of contact? Justify your answer. 1 3. What is the distance between two parallel tangents of a circle of radius 7 cm? 1 4. In the given gure, nd QSR. 1 5. In the given gure, AB, AC and AD are tangents. If AB = 5 cm, nd AD. 1 6. A point P is 26 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 24 cm. Find the radius of the circle. 1 7. From a point P, the length of the tangent to a circle is 15 cm and distance of P from the centre of the circle is 17 cm. Then what is the radius of the circle? 1 8. In gure if ATO = 40 , nd AOB. 2 9. In gure, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm, and BC = 7 cm, then nd the length of BR. 2 10. In gure, ABC is circumscribing a circle. Find the length of BC. 2 11. In gure, there are two concentric circles, with centre O and of radii 5 cm and 3 cm. From an 2 external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, nd the length of BP. 12. In gure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD. 2 13. In gure, O is the centre of the circle, PQ is a tangent to the circle at A. If PAB = 58 , nd ABQ and AQB. 2 14. In gure, a circle touches the side BC of ABC at P and touches AB and AC produced at Q and R respectively. If AQ = 5 cm, nd the perimeter of ABC. 2 15. In the given gure, TAS is a tangent to the circle, with centre O, at the point A. If OBA = 32 , nd the value of x. 2 16. In the given gure, RS is the tangent to the circle at L and MN is a diameter. If NML = 30 , determine RLM. 2 17. Two tangents PA and PB are drawn to the circle with centre O, such that APB = 120 . Prove that 2 OP = 2AP. 18. In the gure, ABCD is a cyclic quadrilateral and PQ is tangent to the circle at C. If BD is a diameter, DCQ = 40 and ABD = 60 , nd BCP. 2 19. The two tangents from an external point P to a circle with centre O are PA and PB. If APB = 70 , 2 what is the value of AOB? 20. In gure, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and CAB = 30 , nd PCA. 2 21. In gure, AOB is a diameter of a circle with centre O and AC is a tangent to the circle at A. If BOC = 130 , then nd ACO. 2 22. In gure, PA and PB are tangents to the circle with centre O such that APB = 50 . Write the measure of OAB. 2 23. In gure, PQ is a chord of a circle with centre O and PT is a tangent. If QPT = 60 , nd PRQ. 2 24. Two concentric circles of radii a and b(a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. 2 25. If d1, d2 (d2 > d1) be the diameters of two concentric circles and c be the length of a chord of a 2 circle which is tangent to the other circle, prove that . 26. In gure, two circles touch each other at the point C. Prove that the common tangent to the circles at C, bisects the common tangent at P and Q. 2 27. In gure, XP and XQ are two tangents to a circle with centre O from a point X outside the circle. ARB is tangent to circle at R. Prove that XA + AR = XB + BR. 2 28. In gure, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and APB = 60 . Find the length of chord AB. 2 29. In gure, common tangents AB and CD to the two circles with centres O1 and O2 intersect at E. Prove that AB = CD. 2 30. In gure, AB is the diameter of a circle with centre O and AT is a tangent. If AOQ = 58 , nd ATQ. 2 31. In gure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If PRQ = 120 , then prove that OR = PR + RQ. 2 32. In gure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. 2 If TPQ = 70 , nd TRQ. 33. ABC is an isosceles triangle, in which AB = AC, circumscribed about a circle. Show that BC is bisected at the point of contact. 3 34. 3 35. In the gure, a circle is inscribed in a quadrilateral ABCD in which B = 90 . If AD = 23 cm, AB = 29 cm and DS = 5 cm, nd the radius (r) of the circle. In gure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D are the lengths 4 cm and 3 cm respectively. If area of ABC = 21 cm2, then nd the lengths of sides AB and AC. 3 36. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre. 3 37. In gure, a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then nd the lengths of AD, BE and CF. 3 38. A tangent PT is drawn parallel to a chord AB as shown in gure. Prove that APB is an isosceles triangle. 3 39. In gure, from an external point P, two tangents PT and PS are drawn to a circle with centre O and 3 radius r. If OP = 2r, show that OTS = OST = 30 . 40. In gure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments 3 BD and DC are respectively of lengths 6 cm and 9 cm. If the area of ABC is 54 cm2, then nd the lengths of sides AB and AC. 41. AB is diameter and AC is a chord of a circle such that BAC = 30 . If tangent at C intersects AB produced in D, prove that BC = BD. 3 42. In a right triangle ABC, a circle with a side AB as diameter is drawn to intersect the hypotenuse AC at P. Prove that the tangent to the circle at P bisects the side BC. 3 43. A quadrilateral ABCD is drawn so that D = 90 , BC = 38 cm and CD = 25 cm. A circle is inscribed 3 in the quadrilateral and it touches the side AB, BC, CD and DA at P, Q, R and S respectively. If BP = 27 cm, nd the radius of the inscribed circle. 44. In gure, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that RPQ = 30 . A chord RS is drawn parallel to the tangent PQ. Find RQS. 3 45. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc. 3 46. In gure, two equal circles, with centres O and O , touch each other at X. OO produced meets the circle with centre O at A. AC is tangent to the circle with centre O, at the point C. O D is 3 perpendicular to AC. Find the value of . 47. In the gure, AB is diameter of a circle with centre O and QC is a tangent to the circle at C. If CAB = 30 , nd CQA and CBA. 4 48. In gure, the sides AB, BC and CA of triangle ABC touch a circle withcentre O and radius r at P, Q and R respectively. Prove that (i) AB + CQ = AC + BQ (ii) Area ( ABC) = 4 (perimeter of ABC) r 49. In gure, PA and PB are two tangents drawn from an external point P to a circle with centre O. Prove that OP is the right bisector of line segment AB. 4 50. PQR is a right angled triangle right angled at Q. PQ = 5 cm, QR = 12 cm. A circle with centre O is inscribed in PQR, touching its all sides. Find the radius of the circle. 4 51. AB is a chord of length 24 cm of a circle of radius 13 cm. The tangents at A and B intersect at a point C. Find the length AC. 4 52. If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + FA. 4 53. Let s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, prove that BD = s b. 4 54. In gure, 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 lengths of TP and TQ. 4 55. In gure, OP is equal to diameter of the circle. Prove that APB is an equilateral triangle. 4 56. A circle is inscribed in a ABC having sides 16 cm, 20 cm and 24 cm as shown in gure. Find AD, 4 BE and CF. 57. PAQ is a tangent to the circle with centre O at a point A as shown in gure. If OBA = 35 , nd the 4 value of BAQ and ACB. 58. O is the centre of a circle. PA and PB are tangents to the circle from a point P. Prove that (i) quadrilateral PAOB is a cyclic quadrilateral (ii) PO is the bisector of APB (iii) OAB = OPA. 4 59. In gure, PQ is a chord of length 16 cm, of a circle of radius 10 cm. The tangents at P and Q intersect at a point T. Find the length of TP. 4 60. In gure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If PBT = 30 , prove that BA : AT = 2 : 1. 4 61. In gure, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT 4 intersects circle at E. If AB is a tangent to the circle at E, nd the length of AB, where TP and TQ are two tangents to the circle.

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Additional Info : CBSE Class 10 : Mathematics : Chap. 10 Circles

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