Circles

Page 118 Circle Chap 10
CHAPTER 10
Circle
OBJECTIVE QUESTION (c) 4.8 cm (d) 6.4 cm

Sol :
5. In figure, O is the centre of circle. PQ is a chord and

1. From an external point Q , the length of tangent to a PT is tangent at P which makes an angle of 50c with
circle is 12 cm and the distance of Q from the centre PQ+POQ is
of circle is 13 cm. The radius of circle (in cm) is
(a) 10 (b) 5
(c) 12 (d) 7
Sol : [Board 2020 Delhi Basic]
2. QP is a tangent to a circle with centre O at a point

P on the circle. If TOPQ is isosceles, then +OQR
equals.
(a) 30c (b) 45c
(c) 60c (d) 90c
(a) 130c (b) 90c
3. A chord of a circle of radius 10 cm, subtends a right (c) 100c (d) 75c
angle at its centre. The length of the chord (in cm) is Sol : [Board 2020 OD Basic]
(a) 5 (b) 5 2
2 6. In figure, AP , AQ and BC are tangents of the circle
(c) 10 2 (d) 10 3
Sol : [Board 2020 OD Basic]
with centre O . If AB = 5 cm , AC = 6 cm and BC = 4
cm, then the length of AP (in cm) is
4. Two chords AB and CD of a circle intersect at E such (a) 15 (b) 10

that AE = 2.4 cm , BE = 3.2 cm and CE = 1.6 cm . (c) 9 (d) 7.5
The length of DE is Sol : [Board 2020 Delhi Basic]
(a) 1.6 cm (b) 3.2 cm
Chap 10 Circle Page 119
7. If a regular hexagon is inscribed in a circle of radius r, (c) 4 cm (d) 6 cm

then its perimeter is Sol :
(a) 3r (b) 6r
(c) 9r (d) 12r 12. In the adjoining figure, PT is a tangent at point C
Sol : of the circle. O is the circumference of TABC . If
+ACP = 118c, then the measure of +x is
8. In figure, on a circle of radius 7 cm, tangent PT is
drawn from a point P such that PT = 24 cm. If O is
the centre of the circle, then the length of PR is
(a) 30 cm (b) 28 cm
(c) 32 cm (d) 25 cm
(a) 28c (b) 32c
(c) 42c (d) 38c
9. Two circles of radii 20 cm and 37 cm intersect in A Sol :
and B . If O1 and O2 are their centres and AB = 24 cm,
then the distance O1 O2 is equal to
(a) 44 cm (b) 51 cm 13. Two concentric circles of radii a and b where a > b,
The length of a chord of the larger circle which touches
(c) 40.5 cm (d) 45 cm the other circle is
Sol :
(a) a2 + b2 (b) 2 a2 + b2
10. In the adjoining figure, TP and TQ are the two (c) a2 - b2 (d) 2 a2 - b2
Sol :
tangents to a circle with centre O . If +POQ = 110c,
then +PTQ is
14. In the given figure, a circle touches all the four sides
of quadrilateral ABCD with AB = 6 cm , BC = 7 cm
and CD = 4 cm , then length of AD is
(a) 60c (b) 70c

(c) 80c (d) 90c
Sol : (a) 3 cm (b) 4 cm
(c) 5 cm (d) 6 cm
11. AB and CD are two common tangents to circles Sol :
which touch each other at a point C . If D lies on AB
such that CD = 4 cm then AB is
(a) 12 cm (b) 8 cm
15. Two concentric circles are of radii 10 cm and 8 cm, PR = 8 cm and QR = 12 cm , then the diameter of the
then the length of the chord of the larger circle which largest circle is:
touches the smaller circle is
(a) 6 cm (b) 12 cm
(c) 18 cm (d) 9 cm
Sol :
16. In the given figure, PA is a tangent from an external

point P to a circle with centre O . If +POB = 115c,
then perimeter of +APO is
(a) 30 cm (b) 20 cm
(c) 10 cm (d) None of these
Sol :
20. If radii of two concentric circles are 4 cm and 5 cm ,

then the length of each of one circle which is tangent
to the other circle, is
(a) 25c (b) 20c (a) 3 cm (b) 6 cm
(c) 30c (d) 65c (c) 9 cm (d) 1 cm
Sol :
21. In figure, if +AOB = 125c, then +COD is equal to

17. From an external point P , tangents PA and PB are
drawn to a circle with centre O . If CD is the tangent
to the circle at a point E and PA = 14 cm . The
perimeter of TPCD is
(a) 14 cm (b) 21 cm
(c) 28 cm (d) 35 cm
Sol :
18. In the given figure, two tangents AB and AC are drawn

to a circle with centre O such that +BAC = 120c,
then OA is equal to that
(a) 62.5c (b) 45c
(c) 35c (d) 55c
Sol :
22. In figure, AT is a tangent to the circle with centre O

such that OT = 4 cm and +OTA = 30c. Then, AT
is equal to
(a) 2AB (b) 3AB

(c) 4AB (d) 5AB
19. In the given figure, three circles with centres P , Q

and R are drawn, such that the circles with centres
Q and R touch each other externally and they touch
the circle with centre P , internally. If PQ = 10 cm ,
(a) 4 cm (b) 2 cm 28. In the adjoining figure, TP and TQ are the two
(c) 2 3 cm (d) 4 3 cm tangents to a circle with centre O . If +POQ = 110c,
Sol : then find the angle +PTQ .
ONE MARK QUESTION
23. From an external point Q , the length of tangent to a

circle is 12 cm and the distance of Q from the centre
of circle is 13 cm. What is the radius of circle?
Sol : [Board 2020 Delhi Basic] Sol : [Board Term-2 Compt. 2016]
24. QP is a tangent to a circle with centre O at a point P 29. In figure, AP , AQ and BC are tangents of the circle
on the circle. If TOPQ is isosceles, then find +OQR with centre O . If AB = 5 cm , AC = 6 cm and BC = 4
? cm, then what is the length of AP ?
25. A chord of a circle of radius 10 cm, subtends a right

angle at its centre. What is the length of the chord?
26. In figure, on a circle of radius 7 cm, tangent PT is

drawn from a point P such that PT = 24 cm. If O
is the centre of the circle, then what is the length of
PR ?
30. Two chords AB and CD of a circle intersect at E such

that AE = 2.4 cm , BE = 3.2 cm and CE = 1.6 cm .
What is the length of DE ?
Sol : [Board Term-2 OD 2011]
Sol : [Board 2020 Delhi Basic] 31. Two circles of radii 20 cm and 37 cm intersect in A
and B . If O1 and O2 are their centres and AB = 24 cm,
27. In figure, O is the centre of circle. PQ is a chord and then find the distance O1 O2 .
Sol : [Board Term-2 Foreign 2016]
PT is tangent at P which makes an angle of 50c with
PQ . Find the angle +POQ .
32. In the figure, QR is a common tangent to given circle
which meet at T . Tangent at T meets QR at P . If
QP = 3.8 cm, then find length of QR .
Sol : [Board Term-2 Delhi 2012, 2014]

33. Two concentric circles of radii a and b where a > b, 39. In given figure, if +AOB = 125c, then find the angle
Find the length of a chord of the larger circle which +COD ?
touches the other circle.
34. If the angle between two radii of a circle is 130c, then

what is the angle between the tangents at the end
points of radii at their point of intersection ?
Sol : [Board Term-2 2012]
35. In the adjoining figure, PT is a tangent at point C

of the circle. O is the circumference of TABC . If
+ACP = 118c, then find the angle +x ?
Sol : [Board Term-2 Delhi Compt. 2014]
40. In the given figure, a circle touches all the four sides
of quadrilateral ABCD with AB = 6 cm , BC = 7 cm
and CD = 4 cm , then what is the length of AD ?

41. To draw a pair of tangents to a circle which are

36. If a circle can be inscribed in a parallelogram how will
inclined to each other at an angle of 30c, it is required
the parallelogram change?
Sol : [Board Term-2, 2014] to draw tangents at end points of two radii of the
circle, what will be the angle between them ?
37. Two concentric circles are of radii 10 cm and 8 cm,
then find the length of the chord of the larger circle
42. In figure, AT is a tangent to the circle with centre O
which touches the smaller circle.
Sol : [Board Term-2 Delhi 2011] such that OT = 4 cm and +OTA = 30c. What is the
length of AT ?
38. In given figure, the length PB = ................... cm.
Sol : [Board Term-2 Delhi 2013]
Sol : [Board 2020 OD Standard]

43. What is the maximum number of parallel tangents a 49. In the given figure, PQ and PR are tangents to the
circle can have on a diameter? circle with centre O such that +QPR = 50º, Then
find +OQR.
44. In the given figure, AOB is a diameter of the circle

with centre O and AC is a tangent to the circle at A
. If +BOC = 130° , the find +ACO.
50. In the figure, PA and PB are tangents to a circle with

centre O . If +AOB = 120c, then find +OPA .
45. Two concentric circles are of radii 5 cm and 3 cm.

Find the length of the chord of larger circle (in cm)
which touches the smaller circle.
Sol : [Board 2020 OD Basic, Foreign 2014]
46. In figure, PA and PB are tangents to the circle with

centre O such that +APB = 50º. Write the measure
of +OAB.
51. What is the length of the tangent drawn from a point

8 cm away from the centre of a circle of radius 6 cm ?
Sol : [Board Term-2, 2012]
52. In figure, PA and PB are two tangents drawn from

an external point P to a circle with centre C and
Sol : [[Board Term-2 Delhi 2015]
radius 4 cm. If PA = PB , then find the length of
each tangent.
47. If PQ and PR are two tangents to a circle with centre

O . If +QPR = 46º then find +QOR .
Sol : [Board Term-2 Delhi 2014] 53. PA and PB are tangents from point P to the circle
with centre O as shown in figure. At point M , a
48. If a line intersects a circle in two distinct points, what
is it called ?
tangent is drawn cutting PA at K and PB at N . 57. In the given figure, find +QSR .
Prove that KN = AK + BN
Sol : [Board Term-2, 2012] 58. In figure, AP and BP are tangents to a circle with
centre O , such that AP = 5 cm and +APB = 60c.
Find the length of chord AB .
54. PQ is a tangent drawn from an external point P
to a circle with centre O , QOR is the diameter of
the circle. If +POR = 120c, What is the measure of
+OPQ ?
Sol : [Board Term-2 Foreign 2017] Sol : [Board Term-2 Delhi 2016]
55. Two tangents making an angle of 60c between them

are drawn to a circle of radius 3 cm, then find the
length of each tangent.
Sol : [Board, Term-2, 2013]
56. In the figure there are two concentric circles with

centre O . PRT and PQS are tangents to the inner
circle from a point P lying on the outer circle. If
PR = 5 cm find the length of PS .
59. In given figure, if AT is a tangent to the circle with

centre O , such that OT = 4 cm and +OTA = 30c,
then find the length of AT .

60. In figure, O is the centre of the circle, PQ is a chord 64. In the given figure, from a point P , two tangents PT
and PT is tangent to the circle at P . Find +OPQ and PS are drawn to a circle with centre O such that
and +TPQ +SPT = 120c, Prove that OP = 2PS .
Sol : [Board Term-2 OD 2017] Sol : [Board Term-2 Foreign 2016]
TWO MARKS QUESTION 65. From an external point P , tangents PA and PB are
drawn to a circle with centre O. If +PAB = 50º, then
find +AOB.
61. A circle is inscribed in a TABC touching AB , BC
and AC at P , Q and R respectively. If AB = 10 cm
AR = 7 cm and CR = 5 cm , then find the length of 66. In given figure, AB is the diameter of a circle with
BC centre O and AT is a tangent. If +AOQ = 58c, find
Sol : [Board 2020 OD Basic] +ATQ .
62. Prove that in two concentric circles, the chord of

the larger circle, which touches the smaller circle is
bisected at the point of contact.
63. In figure, a circle touches all the four sides of a

quadrilateral ABCD . If AB = 6 cm, BC = 9 cm and
CD = 8 cm, then find the length of AD .
67. Prove that the lengths of two tangents drawn from an

external point to a circle are equal.
Sol : [Board 2020 OD Basic, 2018]
68. In the given figure PQ is chord of length 6 cm of the

circle of radius 6 cm. TP and TQ are tangents to the
circle at points P and Q respectively. Find +PTQ .

69. In figure, two tangents RQ and RP are drawn from 73. In Figure, common tangents AB and CD to the two
an external point R to the circle with centre O . If circle with centres O1 and O2 intersect at E. Prove
+PRQ = 120c, then prove that OR = PR + RQ . that AB = CD.

70. In figure, O is the centre of a circle. PT are tangents to
the circle from an external point P . If +TPQ = 70c,
find +TRQ . 74. Two tangents PA and PB are drawn from an external
point P to a circle inclined to each other at an angle
of 70º, then what is the value of +PAB ?
75. In the given figure, BOA is a diameter of a circle and

the tangent at a point P meets BA when produced at
T. If +PBO = 30º , what is the measure of +PTA ?
71. Prove that tangents drawn at the ends of a chord of a

circle make equal angles with the chord.
76. In figure, O is the centre of the circle and LN is a

diameter. If PQ is a tangent to the circle at K and
+KLN = 30º , find +PKL.
72. In Figure a quadrilateral ABCD is drawn to

circumscribe a circle, with centre O, in such a way
that the sides AB, BC, CD, and DA touch the circle
at the points P, Q, R and S respectively. Prove that.
AB + CD = BC + DA.
Sol : [Board Term-2 OD Compt 2017]

77. In the given figure, if AB = AC , prove that BE = CE.
83. In figure, two tangents TP and TQ are drawn to

circle with centre O from an external point T . Prove
that +PTQ = 2+OPQ .
THREE MARKS QUESTION Sol : [Board 2020 Delhi Standard]
84. In given figure, two circles touch each other at the

78. If tangents PA and PB drawn from an external point point C . Prove that the common tangent to the
P to a circle with centre O are inclined to each other circles at C , bisects the common tangent at P and Q .
at an angle of 80c, then find +POA .
79. An isosceles triangle ABC , with AB = AC ,

circumscribes a circle, touching BC at P , AC at Q
and AB at R . Prove that the contact point P bisects
BC .
80. Prove that the rectangle circumscribing a circle is a Sol : [Board 2020 OD Basic, 2020 Delhi Standard]
square.
Sol : [Board 2020 SQP Standard]
85. In the given figure, OP is equal to the diameter of a

81. Prove that the tangent at any point of a circle is circle with centre O and PA and PB are tangents.
perpendicular to the radius through the point of Prove that ABP is an equilateral triangle.
contact.
82. If a circle touches the side BC of a triangle ABC

at P and extended sides AB and AC at Q and R ,
respectively, prove that AQ = 12 (BC + CA + AB)
Sol : [Board 2020 OD Standard, 2016]
86. From a point P , which is at a distant of 13 cm from

the centre O of a circle of radius 5 cm, the pair of
tangents PQ and PR are drawn to the circle, then
the area of the quadrilateral PQOR (in cm2).
87. If O is centre of a circle, PQ is a chord and the 92. Prove that the tangents drawn at the ends of a
tangent PR at P makes an angle of 50c with PQ , diameter of a circle are parallel.
find +POQ . Sol : [Board 2020 Delhi Basic, 2017, 2014]
FIVE MARKS QUESTION
93. In Figure, PQ is a chord of length 8 cm of a circle of

radius 5 cm and centre O . The tangents at P and Q
intersect at point T . Find the length of TP .
88. In the figure, PQ is a tangent to a circle with centre

O . If +OAB = 30º , find +ABP and +AOB.
Sol : [Board 2019 Delhi Standard]
94. If the angle between two tangents drawn from an

external point P to a circle of radius a and centre O,
is 60º, then find the length of OP.
Sol : [Board 2020 SQP STD]

95. A right triangle ABC , right angled at A is
circumscribing a circle. If AB = 6 cm and BC = 10 cm,
89. In TABD, AB = AC. If the interior circle of TABC find the radius r of the circle.
touches the sides AB, BC and CA at D, E and F Sol : [Board 2020 Delhi Basic]
respectively. Prove that E bisects BC.
96. In figure, a circle is inscribed in a TABC having sides
BC = 8 cm , AB = 10 cm and AC = 12 cm . Find the
90. A circle is inscribed in a TABC, with sides AC, AB length BL, CM and AN .
and BC as 8 cm, 10 cm and 12 cm respectively. Find
the length of AD, BE and CF.
91. In the given figure, PA and PB are tangents to a

circle from an external point P such that PA = 4 cm
and +BAC = 135º . Find the length of chord AB .
Sol : [Board 2019 Delhi]

97. a, b and c are the sides of a right triangle, where c 100. In figure, a circle with centre O is inscribed in a
is the hypotenuse. A circle, of radius r , touches the quadrilateral ABCD such that, it touches the sides
sides of the triangle. Prove that r = a + b − c . BC , AB, AD and CD at points P, Q, R and S
Sol : 2 Term-2 Delhi 2016]
[Board respectively. If AB = 29 cm, AD = 23 cm, +B = 90c
and DS = 5 cm, then find the radius of the circle.
98. PB is a tangent to the circle with centre O to B.AB
is a chord of length 24 cm at a distance of 5 cm from
the centre. It the tangent is length 20 cm, find the
length of PO .
101. Prove that opposite sides of a quadrilateral

circumscribing a circle subtend supplementary angles
at the centre of the circle.
Sol : [Board Term-2 Foreign 2017, OD 2014]
102. Prove that tangent drawn at any point of a circle

perpendicular to the radius through the point contact.
Sol : [Board Term-2 Delhi 2015] Sol : [Board Term-2 OD 2016]
99. In figure O is the centre of a circle of radius 5 cm. T 103. Two tangents PA and PB are drawn from an
is a point such that OT = 13 cm and OT intersects external point P to a circle with centre O , such that
circle at E . If AB is a tangent to the circle at E , find +APB = +x and +AOB = y . Prove that opposite
the length of AB , where TP and TQ are two tangents angles are supplementary.
to the circle. Sol : [Board Term-2, 2011]
104. In figure, PQ is a chord of a circle O and PT is a

tangent. If +QPT = 60c, find +PRQ .
Sol : [Board Term-2 OD 2015, 2017]
105. In Figure the radius of incircle of TABC of area

84 cm2 and the lengths of the segments AP and BP
into which side AB is divided by the point of contact
are 6 cm and 8 cm Find the lengths of the sides AC
Sol : [Board Term-2 Delhi 2016] and BC.
Sol : [Board Term-2 Delhi 2012, 2014, OD Compt. 2017]

106. In the given figure, O is the centre of the circle. 109. In Figure, PQ and RS are two parallel tangents to
Determine +APC , if DA and DC are tangents and a circle with centre O and another tangent AB with
+ADC = 50c. point of contact C intersecting PQ at A and RS at
B . Prove that +AOB = 90c.
Sol : [Board 2019 OD STD, 2014, 2012]
110. Prove that the parallelogram circumscribing a circle

107. AB is a chord of circle with centre O . At B , a tangent is a rhombus.
Sol : [Board 2020 Delhi STD, 2013, 2014]
PB is drawn such that its length is 24 cm. The
distance of P from the centre is 26 cm. If the chord
AB is 16 cm, find its distance from the centre. 111. In given figure, PA and PB are tangents from a point
P to the circle with centre O . At the point M , other
tangent to the circle is drawn cutting PA and PB at
K and N . Prove that the perimeter of TPNK = 2PB
.

108. From a point T outside a circle of centre O , tangents

112. In the figure, tangents PQ and PR are drawn from an
TP and TQ are drawn to the circle. Prove that OT is
external point P to a circle with centre O , such that
the right bisector of line segment PQ .
Sol : [Board Term-2 Delhi 2015] +RPQ = 30c. A chord RS is drawn parallel to the
tangent PQ . Find +RQS .

113. In the figureAB and CD are common tangents to two 116. In the given figure, PA and PB are tangents to
circles of unequal radii. Prove that AB = CD . the circle from an external point P . CD is another
tangent touching the circle at Q . If PA = 12 cm,
QC = QD = 3 cm, then find PC + PD.
114. In figure PQ is a tangent from an external point P Sol : [Board Term-2 Delhi Compt. 2017]
to a circle with centre O and OP cuts the circle at T
and +QOR is a diameter. If +POR = 130c and S is COMPETENCEY BASED QUESTION
a point on the circle, find +1 + +2 .
117. A children’s park is in the triangular shape as shown

in Figure below. In the middle of the park, there is
a circular region for younger children to play. It is
fenced with three layers of wire. The radius of the
circular region is 3 m.
(i) Find the perimeter (or circumference) of the

circular region.
(ii) What is the total length of wire used?
115. In the given figure, AD is a diameter of a circle with (iii) What is the area of the circular region?
centre O and AB is a tangent at A. C is a point (iv) If BD = 6 m , DC = 9 m and ar (∆ABC) = 54 m2
on the circle such that DC produced intersects the , then find the length of sides AB and AC ,
tangent at B and +ABC = 50c. Find +AOC . respectively.
(v) Find the perimeter of ∆ABC .
Sol : [Board 2021 Comp. Basic]

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Page 118 Circle Chap 10

OBJECTIVE QUESTION (c) 4.8 cm (d) 6.4 cm

5. In figure, O is the centre of circle. PQ is a chord and

2. QP is a tangent to a circle with centre O at a point

4. Two chords AB and CD of a circle intersect at E such (a) 15 (b) 10

7. If a regular hexagon is inscribed in a circle of radius r, (c) 4 cm (d) 6 cm

(a) 60c (b) 70c

16. In the given figure, PA is a tangent from an external

20. If radii of two concentric circles are 4 cm and 5 cm ,

21. In figure, if +AOB = 125c, then +COD is equal to

18. In the given figure, two tangents AB and AC are drawn

22. In figure, AT is a tangent to the circle with centre O

(a) 2AB (b) 3AB

19. In the given figure, three circles with centres P , Q

ONE MARK QUESTION

23. From an external point Q , the length of tangent to a

25. A chord of a circle of radius 10 cm, subtends a right

26. In figure, on a circle of radius 7 cm, tangent PT is

30. Two chords AB and CD of a circle intersect at E such

Sol : [Board 2020 OD Basic]

Sol : [Board Term-2 Delhi 2012, 2014]

34. If the angle between two radii of a circle is 130c, then

35. In the adjoining figure, PT is a tangent at point C

Sol : [Board Term-2 OD 2017]

41. To draw a pair of tangents to a circle which are

Sol : [Board Term-2 Delhi 2013]

Sol : [Board 2020 OD Standard]

44. In the given figure, AOB is a diameter of the circle

Sol : [Board Term-2 Delhi 2012, 2015]

50. In the figure, PA and PB are tangents to a circle with

45. Two concentric circles are of radii 5 cm and 3 cm.

46. In figure, PA and PB are tangents to the circle with

51. What is the length of the tangent drawn from a point

52. In figure, PA and PB are two tangents drawn from

47. If PQ and PR are two tangents to a circle with centre

Sol : [Board Term-2, 2013]

Sol : [Board Term-2, 2012]

55. Two tangents making an angle of 60c between them

56. In the figure there are two concentric circles with

59. In given figure, if AT is a tangent to the circle with

Sol : [Board Term-2 Delhi Compt. 2017]

Sol : [Board Term-2, 2012]

Sol : [Board Term-2 OD 2017] Sol : [Board Term-2 Foreign 2016]

62. Prove that in two concentric circles, the chord of

63. In figure, a circle touches all the four sides of a

67. Prove that the lengths of two tangents drawn from an

68. In the given figure PQ is chord of length 6 cm of the

Sol : [Board Term-2 Delhi 2016]

Sol : [Board Term-2 OD 2015]

75. In the given figure, BOA is a diameter of a circle and

71. Prove that tangents drawn at the ends of a chord of a

Sol : [Board Term-2, 2012]

76. In figure, O is the centre of the circle and LN is a

72. In Figure a quadrilateral ABCD is drawn to

Sol : [Board Term-2 OD Compt 2017]

Sol : [Board Term-2 OD 2016]

77. In the given figure, if AB = AC , prove that BE = CE.

83. In figure, two tangents TP and TQ are drawn to

Sol : [Board Term-2 OD 2017]