6) circles - questions

1. Prove that the lengths of tangents drawn from anexternal point to a circle are equal. Using the above,prove the followingA quadrilateral ABCD is drawn to circumscribe a circle.Prove that AB + CD = AD + BC.

2. Prove that the lengths of the tangents drawn from anexternal point to a circle are equal. Using the above, dothe following:In the fig., TP and TQ are tangents from T to the circlewith centre O and R is any point on the circle.If AB is a tangent to the circle at R, prove that TA + AR =TB + BR.

3. Prove that the lengths of tangents drawn from anexternal point to a circle are equal. Using the above dothe following :ABC is an isosceles triangle in which AB = AC,circumscribe about a circle as shown in the fig. Provethat the base is bisected by the point of contact.

4. Prove that the tangent at any point of a circle isperpendicular to the radius through the point ofcontact. Using the above, do the following:In fig., O is the centre of the two concentric circles. AB isa chord of the larger circle touching the small circle at C.Prove that AC = BC.

5. Prove that the length of the tangents drawn from anexternal point to a circle are equal. Using the above, dothe following :In fig, quadrilateral ABCD is circumscribing a circle. Findthe perimeter of the quadrilateral ABCD.

6. If ABC is isosceles with AB = AC, prove that thetangent at A to the circumcircle of ABC is parallel toBC.

7. In figure, AB and CD are two parallel tangents to a circlewith centre O. ST is tangent segment between the twoparallel tangents touching the circle at Q. Show that

SOT = 900.

8. A circle is inscribed in a ABC having sides 8 cm, 10cm and 12 cm as shown in figure. Find AD, BE and CF.

9. PAQ is a tangent to the circle with centre O at a point Aas shown in figure. If OBA = 350, find the value of

BAQ and ACB.

10. AB is diameter and AC is a chord of a circle such thatBAC = 300. If then tangent at C intersects AB

produced in D, prove that BC = BD.

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11. ABC is an isosceles triangle in which AB = AC,circumscribed about a circle. Show that BC is bisectedat the point of contact.ORIn the fig., a circle is inscribed in a quadrilateral ABCD inwhich B = 900. If AD = 23 cm, AB = 29 cm and DS = 5cm, find the radius (r) of the circle.

12. In fig., OP is equal to diameter of the circle. Prove thatABP is an equilateral triangle.

13. Prove that a parallelogram circumscribing a circle is arhombus.

14. Two tangents PA and PB are drawn to a circle withcentre O from and external point P. Prove that APB= 2 OAB.

15. In fig., a circle is inscribed in a triangle ABC having sideBC = 8 cm, AC = 10 cm and AB = 12 cm. Find AD, BE andCF

16. In fig., there are two concentric circles with centre Oand of radii 5 cm and 3 cm. From an external Point P,tangents PA and PB are drawn to these circles. If AP =12 cm, find the length of BP.

17. In the given figure, TAS is a tangent to the circle, withcentre O, at the point A. If OBA = 320, find the valueof x.

18. In the given figure, ABC is a right angled triangle rightangled at A, with AB = 6cm and AC = 8 cm. A circle withcentre O has been inscribed inside the triangle.Calculate the value of r, the radius of the inscribedcircle.

19. In the given figure, PT is tangent to the circle at T. If PA= 4 cm and AB = 5 cm, find PT.


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20. In the figure, O is the centre of the circle, PQ is tangentto the circle at A. If PAB = 580, find ABQ and

AQB.

21. In figure, a circle touches the side BC of ABC at P andtouches AB and AC produced at Q and R respectively. IfAQ = 5 cm, find the perimeter of ABC.

22. A tangent PT is drawn parallel to a chord AB as shown infigure. Prove that APB is an isosceles triangle.

23. In figure, XP and XQ are two tangents to a circle withcentre O from a point X outside the circle. ARB istangent to circle at R. Prove that XA + AR = XB + BR.

24. In fig, if ATO = 400, find AOB.

25. In fig., 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 find the length of BR.

26. In fig., BC is circumscribing a circle. Find the length ofBC.

27. In fig., CP and CQ are tangents from an external point Cto a circle with centre O. AB is another tangent whichtouches the circle at R. If CP = 11 cm and BR = 4 cm, findthe length of BC.

28. ABC is right-angled triangle with AB = 12 cm and AC =13 cm. A circle with centre O has been inscribed insidethe triangle. Calculate the value of x, the radius of theinscribed circle.

29. PQR is a right-angled triangle with PQ = 3 cm and QR = 4cm. A circle which touches all the sides of the triangle isinscribed in the triangle. Calculate the radius of thecircle.


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30. In the given figure, O is the centre of each one of twoconcentric circles of radii 4 cm and 6 cm respectively. PAand PB are tangents to outer and inner circlerespectively. If PA = 10 cm, find the length of PB, up totwo places of decimal.

31. In the given figure, ABC is circumscribed. The circletouches the sides AB, BC and CA at P, Q, R respectively.If AP = 5 cm, BP = 7 cm, AC = 14 cm and BC = x cm, findthe value of x.

32. In the given figure, quadrilateral ABCD is circumscribed.The circle touches the sides AB, BC, CD and DA at P, Q,R, S respectively. If AP = 9 cm, BP = 7 cm, CQ = 5 cm andDR = 6 cm, find the perimeter of quad. ABCD.

33. In the given figure, the circle touches the sides AB, BC,CD and DA of a quadrilateral ABCD at the points P, Q, R,Srespectively. If AB = 11 cm, BC = x cm, CR = 4 cm and AS= 6 cm, find the value of x.

34. In the given figure, a circle touches the side BC ofABC at P and AB and AC produced at Q and R

respectively. If AQ = 15 cm, find the perimeter ofABC.

35. In the given figure, PA and PB are two tangents to thecircle with centre O. If APB = 400, find AQBand AMB.

36. In the given figure, PA and PB are two tangents to thecircle with centre O. If APB =500, find:(i) AOB (ii) OAB (iii) ACB

37. In the given figure PQ is a diameter of a circle withcentre O and PT is a tangent at P. QT meets the circle atR. If POR = 720, find PTR.


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38. In the given figure, O is the centre of the circumcircle ofABC. Tangents at A and B intersect at T.

If ATB =800 and AOC = 1300, Calculate CAB.

39. In the given figure, PA and PB are tangents to a circlewith centre O and ABC has been inscribed in thecircle such that AB = AC. If BAC = 720, calculate (a)

AOB (B0 APB.

40. Show that the tangent lines at the end points of adiameter of a circle are parallel.

41. Prove that the tangents at the extremities of any chordmake equal angles with the chord.

42. Show that the line segment joining the points of contactof two parallel tangents passes through the centre.

43. In the given figure, PQ is a transverse common tangentto two circles with centres A and B and of radil 5 cm and3 cm respectively If PQ intersects AB at C such that CP =12 cm, calculate AB.

44. ABC is an isosceles triangle in which AB = AC,circumscribed about a circle. Prove that the base isbisected by the point of contact.

45. In the given figure quadrilateral ABCD is circumscribedand AD AB. If the radius of incircle is 10 cm,find the value of x.

46. In the given figure, a circle is inscribed in quad. ABCD. IfBC = 38 cm, BQ = 27 cm, DC = 25 cm andAD DC, find the radius of the circle.


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47. Find the length of the tangent drawn to a circle of radius8 cm, from a point which is at a distance of 10 cm fromthe centre of the circle.

48. A point P is 7 cm away from the centre of the circle andthe length of tangent drawn from P to the circle is 15cm. Find the radius of the circle.

49. There are two concentric circles, each with centre O andof radii 10 cm and 26 cm respectively. Find the length ofthe chord AB of the outer circle which touches the innercircle at P.

50. A and B are centres of circles of radii 9 cm and 2 cm suchthat AB = 17 cm and C is the centre of the circle of radiusr cm which touches the above circles externally. If

ACB = 90o, write an equation in r and solve it.

51. Two circles touch each other extemally at a point C andP is a point on the common tangent at C. If PA and PBare tangents to the two circles, prove that PA = PB.

52. Two circles touch each other internally. Prove that thetangents drawn to the two circles from any point on thecommon tangent are equal in length.

53. Two circles of radii 18 cm and 8 cm touch externally.Find the length of a direct common tangent to the twocircles.

54. Two circles of radii 8 cm and 3 cm have their centres 13cm apart. Find the length of a direct common tangent tothe two circles.

55. Two circles of radii 8 cm and 3 cm have a direct commontangent of length 10 cm. Find the distance betweentheir centres, upto two places of decimal.

56. With the vertices of PQR as centres, three circles aredescribed, each touching the other two externally. If thesides of the triangle are 7 cm, 8 cm and 11 cm, find theradii of the three circles

57. In the given figure, PA and PB are the tangent segmentsto a circle with centre O. Show that the points A, O, Band P are concyclic.

58. From an external point P, tangents PA and PB are drawnto a circle with centre O. If CD is the tangent to the circleat a point E and PA = 14cm, find the perimeter of D PCD.

59. A circle is inscribed in a DABC having AB = 10 cm, BC = 12cm and CA = 8 cm and touching these sides at D, E, Frespectively, as shown in the figure. Find AD, BE and CF.


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60. In the given figure, ABCD is a quadrilateral in whichÐ D = 900. A circle C(O, r) touches the sides AB, BC, CDand DA at P, Q, R , S respectively. If BC = 38 cm, CD = 25cm and BP = 27 cm, find the value of r.

61. In the given figure, a circle touches all the four sides of aquadrilateral ABCD whose three sides are AB = 6 cm, BC= 7 cm and CD = 4cm. Find AD.

62. In the given figure, PA and PB are tangents such that PA= 9 cm and ÐAPB = 600. Find the length of chord AB.

63. From a point P, two tangents PA and PB are drawn to acircle C (O, r). If OP = 2r, show that DAPB is equilateral.

64. Find the length of tangent drawn to a circle with radius 7cm from a point 25 cm away from the centre of thecircle.

65. A point P is 26 cm away from the centre of a circle andthe length of the tangent drawn from P to the circle is24 cm. Find the radius of the circle.

66. Two tangent segments BC and BD are drawn to a circlewith centre O such that ÐCBD = 1200 . Prove that OB =2BC.

67. In the given figure, O is the centre of two concentriccircles of radii 4 cm and 6 cm respectively. PA and PB aretangents to the outer and inner circle respectively. If PA= 10 cm, find the length of PB up to one place ofdecimal.

68. ABCD is a quadrilateral such than = 900. A circle C(O, r) touches the sides AB, BC, CD and DA at P, Q, R andS respectively. If BC = 38 cm, CD = 25 cm and BP = 27 cm,find r.

69. Two concentric circles are of radius 5 cm and 3 cm. Findthe length of the chord of the larger circle whichtouches the smaller circle.

70. In a circle of radius 5 cm, AB and AC are two chords,such that AB = AC = 6 cm. Find the length of chord BC.


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71. The radius of the incircle of a triangle is 4 cm and thesegments into which one side is divided by the point ofcontact are 6 cm and 8 cm. Determine the other twosides of the triangle.

72. In figure, and m are two parallel tangents at P and R.The tangent at Q makes an intercept ST betweenand m. Prove that = 900

73. PQR is a right angled triangle with PQ = 12 cm and QR =5 cm. A circle with centre O and radius x is inscribed in

PQR. Find the value of x.

74. From an external point P, two tangents PA and PB aredrawn to the circle with centre O. Prove that OP is theperpendicular bisector of AB.

75. Two tangents TP and TQ are drawn to a circle withcentre O from an external point T. Prove that

.76. A circle touches the sides of a quadrilateral ABCD at P,

Q, R, S respectively. Show that the angles subtended atthe centre by a pair of opposite sides aresupplementary.

77. In figure, a circle touches all the four sides of aquadrilateral ABCD with AB= 6 cm, BC = 7 cm and CD =4 cm. Find AD.

78. Prove that the lengths of the tangents drawn from anexternal point to a circle are equal.Using the above, dothe following :In figure, TP and TQ are tangents from T to the circlewith centre O and R is any point on the circle. If ABs a tangent to the circle at R, prove thatTA + AR = TB + BR.

79. In figure, if = 400, find

80. In figure OP is equal to diameter of the circle. Prove thatABP is an equilateral triangle.

81. A point A is 26 cm away from the centre of a circle andthe length of tangent drawn from A to the circle is 24cm. Find the radius of the circle.


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82. In the given figure, ABC is right-angled at B, in whichAB = 15 cm and BC = 8 cm. A circle with centre O hasbeen inscribed in ABC. Calculate the value of x, theradius of the inscribed circle.

83. If all the sides of a parallelogram touch a circle, showthat the parallelogram is a rhombus.

84. In the given figure, the in circle of ABC touches thesides AB, BC and CA at the points P, Q, R respectively.

Show that AP + BQ + CR = BP + CQ + AR = (Perimeterof ABC

85. In two concentric circles, prove that a chord of largercircle which is tangent to smaller circle is bisected at thepoint of contact.

86. Two concentric circles are of radii 13 cm and 5 cm. Findthe length of the chord of the outer circle which touchesthe inner circle.

87. In the given figure, PT is a common tangent to the circlestouching externally at P and AB is another commontangent touching the circles at A and B. Prove that:(i) T is the mid-point of AB(ii) APB = 90(iii) If X and Y are centres of the two circles,show that the circle on AB as diameter touches the lineXY.

88. Two circles of radii 25 cm and 9cm touch each otherexternally. Find the length of the direct commontangent.

89. In the given figure, PQ = QR, RQP = 680, PC and CQare tangents to the circle with centre O. Calculate thevalues of : (i) QOP (ii) QCP

90. With the vertices of ABC as centres, three circles aredescribed, each touching the other two externally. If thesides of the triangle are 9 cm, 7 cm and 6 cm, find theradii of the circles.

91. If all the sides of a parallelogram touches a circle, showthat the parallelogram is a rhombus.

92. A circle touches the side BC of a ABC at P and touchesAB and AC when produced at Q and R respectively as

shown in figure, Show that (Perimeter of ABC).93. Prove that the tangents at the extremities of any chord

make equal angles with the chord.94. Prove that the segment joining the points of contact of

two parallel tangents passes through the centre.95. If all the sides of a parallelogram touches a circle,show that the parallelogram is a rhombus96. A circle touches the BC of a ABC at P and touches

AB and AC when produced at Q and R respectively

as shown in figure, Show that (Perimeter ofABC).

97. Prove that the tangents at the extremities of anychord make equal angles with the chord.

98. Prove that the segment joining the points of contactof two parallel tangents passes through the centre.

99. ABCD is a quadrilateral such than = 900. Acircle C (O, r) touches the sides AB, BC, CD and DAat P, Q, R and S respectively. If BC = 38 cm, CD =25 cm and BP = 27 cm, find r.

100. Two concentric circles are of radius 5 cm and 3 cm.Find the length of the chord of the larger circle whichtouches the smaller circle.


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101. In a circle of radius 5 cm, AB and AC are two chords,such that AB = AC = 6 cm. Find the length of chordBC.

102. The radius of the incircle of a triangle is 4 cm and thesegments into which one side is divided by the

point of contact are 6 cm and 8 cm. Determine theother two sides of the triangle.103. PQR is a right angled triangle with PQ = 12 cm and

QR = 5 cm. A circle with centre O and radius x isinscribed in PQR. Find the value of x.

104. In figure, a circle touches all the four sides of aquadrilateral ABCD with AB= 6 cm, BC = 7 cm andCD = 4 cm. Find AD.[CBSE - 2002]

105. In figure, if = 400, find [CBSE - 2008]

106. In figure, and m are two parallel tangents at P andR. The tangent at Q makes an intercept ST between

and m. Prove that = 900

107. From an external point P, two tangents PA and PBare drawn to the circle with centre O. Prove that OPis the perpendicular dissector of AB.

108. Two tangent TP and TQ are drawn to a circle withcentre O from an external point T. Prove

that .

109. A circle touches the sides of a quadrilateral ABCD atP, Q, R, S respectively. Show that the anglessubtended at the centre by a pair of opposite sidesare supplementary.

110. Prove that the lengths of the tangents drawn from anexternal point to a circle are equal.Using the above, do the following :In figure, TP and TQ are tangents from T to the circlewith centre O and R is any point on the circle. If AB isa tangent to the circle at R, prove thatTA + AR = TB + BR. [CBSE - 208]

111. In figure OP is equal to diameter of the circle. Provethat ABP is an equilateral triangle.[CBSE - 2008]

112. Draw a circle of radius 2.5 cm. Take a point P on it.Draw a tangent to the circle at the point P.

113. From a point P on the circle of radius 4 cm, draw atangent to the circle without using the centre. Also,write steps of construction.

114. Draw a circle of radius 3.5 cm. Take a point P on it.Draw a tangent to the circle at the point P, withoutusing the centre of the circle.

115. Draw a circle of radius 3 cm. Take a point P at adistance of 5.6 cm from the centre of the circle. Fromthe point P, draw two tangents to the circle.

116. Draw a circle of radius 4.5 cm. Take point P outsidethe circle. Without using the centre of the circle, drawtwo tangents to the circle from the point P.

117. Construct a triangle ABC, similar to a givenequilateral triangle PQR with side 5 cm. such thateach of its side is 6/7th of the corresponding side ofthe PQR.

118. Construct a triangle ABC. similar to a given isoscelestriangle PQR with QR = 5 cm, PR = PQ = cm, suchthat each of its side is 5/3 of the corresponding sidesof the PQR.


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119. Draw a line segment AB = 7 cm. Divide it externallyin the ratio of(i) 3 : 5 (ii) 5 : 3

120. Draw a ABC with side BC = 6 cm, AB = 5cm and= 600. Construct a AB’C’ similar to ABC

such that sides of AB’C’ are of thecorresponding sides of ABC. [CBSE - 2008]


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6) circles - questions

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