Download 17 Geometry (Circle) (MC)

Mathematics (Multiple Choices)
17 – Geometry (Circle)
HKCEE 1980’
1 24.
○
A.
C.
E.
In the figure, the two circles intersect at A and B. CAE and CBD are straight lines. CED 
y .
180  x  y .
360  x  y .
2 25.
○
B.
D.
180  y .
180  x  y .
In the figure, circle AXB passes through the centre of circle AYB. y =
A.
2x.
B.
180 2x.
C.
180 x.
D.
1
(90  x) .
2
E.
1
(180  x) .
2
3 27.
○
In the figure, the inscribed circle of ABC touches AC at D. If AB = 7, AC = 5 and AD = 2,
then BC =
A.
C.
E.
4 42.
○
A.

.
10
C.
3
.
10
E.
9
.
10
5 47.
○
A.
C.
E.
B.
D.
9.5.
8.5.
7.5.
9.
8.
In the figure, diameter AB = 2. CAB = 18 . Minor arc BC =
B.

.
5
D.
4
.
5
B
C
2

18
10 
A
In the figure, AB = BC = CD. AED =
50 .
75 .
105 .
B.
D.
65 .
90 .
235
HKCEE 1980’
6 48.
○
A.
C.
E.
the chords AB, BC and CA must be equal in length?
AB and BC only
B. AC and BC only
AB and AC only
D. All of them
No two of them
7 49.
○
A.
C.
E.

In the figure, AB = AC, D is the midpoint of B C . Which of the following is/are true?
I. AD bisects BAC
II. BC AD
III. AD is a diameter of the circle
I only
B. II only
III only
D. I and II only
II and III only
8 50.
○
A.
C.
E.
In the figure, RS is a tangent to the circle at C. BA is any chord parallel to RCS. Which of
In the figure, AOB is a diameter of the circle, centre O. CD is the perpendicular bisector of
OA. Which of the angles a, b, c, d is/are equal to 30 ?
a only
B. a and b only
a, b and c only
D. a, b, c and d
None of them
9 51.
○
In the figure, circle O is inscribed in ABC, touching BC at X. Which of the following must
be true?
A.
C.
E.
I. OX BC
II. OA bisect A
III. AO produced bisect BC
I only
B. I and II only
I and III only
D. I, II and III only
None of them
10 54.
○
A.
C.
E.
In the figure, O is the centre of the circle. PAB is a straight line. x + y =
2 .
180   .
180 .
B.
D.
90   .
180  2 .
236
HKCEE 1981’
11 24.
○
A.
B.
C.
D.
E.
In the figure, BE is a diameter of the circle. ABC and EDC are straight lines. xo =
C
o
21 .
31o.
38o.
52o.
59o.
21o
D
xo
38o
B
E
A
12 25.
○
In the figure, AT touches the circle at A. In ABC, A : B : C = 2 : 3 : 4.  =
C
o
A.
B.
40 .
50o.
C.
D.
E.
60o.
70o.
80o.
B
A

T
13 26.
○
In the figure, AB is a diameter of the circle with centre at O. The length of the minor arc AC
is twice the length of the minor arc CD. BOD =
o
A.
B.
C.
72 .
90o.
108o.
D.
E.
132o.
144o.
14 27.
○
24o
B
C
D
In the figure, two circles both with radius 2 cm touch each other externally. AP and AQ are
A
equal tangents to the two circles. AP =?
P
Q
120 o
A.
3 cm.
B.
2 3 cm.
D.
4 3 cm.
E.
4 3
cm.
3
C.
15 51.
○
A.
E.
O
A
4 cm.
In the figure, AB and AC touch the circle at B and C. If P is any point on the minor arc BC,
A
what is  ?
o
56o
112
B. 118o
C. 124o
D. 146o
P
It cannot be determined.
C

B
237
HKCEE 1981’
16 52.
○
I.
A
B
D
II.
C
BAC = 90o
ADBC.
A
B
E
C
D
III.
AC and BC
intersect at E.
A
T
C
B
BC produced meets the
tangent AT at T.
A.
Which of the above figures contains one or more pairs of similar triangles?
I only
B. I and II only C. I and III only D. II and III only E. I, II and III
HKCEE 1982’
17 27.
○
A.
a  b.
B.
a + b.
D.
1
a.
2
E.
1
a + b.
2
C.
D
A
In the figure, AKC and BKD are two chords of the circle. CBD =
b
a + b  90o.
a
E
B
C
18 28.
○
A.
C.
E.
48o.
84o.
96o.
In the figure, PQ and RS touch the circle at A and C respectively. ABC =
B.
D.
P
60o.
90o.
A
78o
Q
B
238
R
18o
C
S
HKCEE 1982’
19 47.
○
A.
B.
C.
D.
E
In the figure, BP is a diameter of the circle. The minor arc AB and the radius are of equal
length. APB =
28.6 . (correct to 3 significant figures)
57.3 . (correct to 3 significant figures)
30 .
45 .
60 .
20 52.
○
B
A
P
In the figure, O is the centre of the circle. PQ is a diameter. Which of the following is/are
true?
Q
I. a = b
II. c = 2a
III. c + d = 180o
A.
D.
I only
II and III only
21 53.
○
B.
E.
I and II only
I, II and III
O
a
C.
I and III only
b
c
P
d
In the figure, the length of the minor arc CD is half the length of the minor arc BC.
A
ACD 
o
A.
30 .
B.
C.
D.
E.
35o.
40o.
45o.
50o.
22 54.
○
D
50o
55o.
40o.
35o.
D.
E.
30o.
20o.
B
C
In the figure, TP and TQ touch the circle at P and Q respectively. R is the point on TQ
T
produced such that PR passes through the centre O of the circle. QPR =
A.
B.
C.
70o
P
O
Q
20o
R
239
HKCEE 1983’
23 24.
○
A.
C.
E.
In the figure, chords AB and CD intersect at P. BP = DP. CAD =
58o.
88o.
142o.
B.
D.
A
C
86o.
92o.
100 o
P
42o
B
24 25.
○
D
In the figure, the three sides of ABC touch the circle at the points P, Q and R. PQR =
A
o
A.
B.
C.
D.
30 .
50o.
55o.
70o.
E.
75o.
30o
Q
P
70o
B
25 53.
○
In the figure, chords AB and CD intersect at E. The length of the minor arc BD is three times
C
the length of the minor arc AC. BAD =
A.
B.
C.
D.
31o.
35o.
42o.
45o.
E.
56o.
124 o
B
A
E
D
26 54.
○
In the figure, PQ and XY touch the circle at A and B respectively. PQ // XY and
QAC  60 . CBX =
A.
B.
C.
D.
E.
C
P
A
P
150 .
135o.
120o.
110o.
100o.
Q
60o
o
C
X
Y
B
HKCEE 1984’
27 24.
○
A.
C.
E.
30o.
50o.
85o.
In the figure, AB and AC touch the circle at B and C respectively. A =
B.
D.
40o.
80o.
B
56o
240
A
C
D
HKCEE 1984’
28 25.
○
In the figure, O is the centre of the circle. TA and TB touch the circle at A and B respectively.

A
OA = 2. The length of the AP B is
A.
D.

.
4
3
2
.
29 53.
○
A.
5.
B.
C.
D.
6.
6.4.
8.
E.
9.6.
B.

.
2
E.
3 .
C.
2
3
.
4
P
O
45o
T
B
In the figure, AB is a diameter of the circle. AP = AQ. AB = 10 and BP = 8. PQ =
P
8
A
B
10
Q
30 54.
○
In the figure, the chords BA and CD, when produced, meet at P. The chords AD and BC,
P
when produced, meet at Q. B =
A.
B.
35o.
40o.
C.
D.
E.
45o.
50o.
55o.
40o
A
30o
B
HKCEE 1985’
31 22.
○
40 .
50 .
65 .
32 25.
○
Q
C
In the figure, ABCD is a cyclic quadrilateral. BA is produced to E. DA bisects CAE .
BCD =
A.
C.
E.
D
B.
D.
45 .
55 .
In the figure, ABCD is a rhombus B is the centre of the circle. ABC =
A. 105 .
C. 130 .
E. 150 .
B.
D.
120 .
135 .
241
HKCEE 1985’
33 48.
○

In the figure, AB is a diameter of the circle ABC. If AC has the same length as AB, then
CAB =
A.
B.
C.
D.
E.
90 .
61.4 . (correct to 3 significant figures)
32.7 . (correct to 3 significant figures)
 24.6 . (correct to 3 significant figures)
151 . (correct to 3 significant figures)
34 53.
○
In the figure, FG touches the circle at E. The chord CB is produced to meet FG at A. ACE
=
A.
C.
E.
10 .
25 .
35
35 54.
○
B.
D.
20
30
In the figure the circle touches the sides of ABC at X, Y and Z. O is the centre of the circle.
Which of the following must be true?
I. OA bisects BAC
II. A, X, O and Z are concyclic
III. AX = AZ
A.
C.
E.
III only
I and III only
I, II and III
B.
D.
I and II only
II and III only
HKCEE 1986’
36 25.
○
In the figure, A, B, C, D and E lie on a circle. AC intersects BE at K. ACD = 100o and
B
CDE = 130 . If BE // CD, then ACB =
25 .
B. 30o.
36o.
D. 40o.
42o.
o
A.
C.
E.
A
o
C
100 o
130 o
E
37 49.
○
A.
D.
D
DA and DC are equal chords of the circle ABCD. CDB = 40o and DAB = 100o. ADB 
C
o
20 .
35o.
B.
E.
o
25 .
40o.
C.
o
30 .
D
242
40o
100 o
A
B
HKCEE 1986’
38 50.
○
A.
C.
E.
In the figure, AB and AC are tangents to the circle BCD. If BDC = 50o, then A =
o
130 .
85o.
50o.
B.
D.
B
o
100 .
80o.
D
50o
A
C
HKCEE 1987’
39 20.
○
In the figure, chords AC and BD meet at E and AB // DC. If CED = 104o, find ABD.
B
C
o
A.
B.
C.
76
52o
38o
D.
E.
14o
It cannot be determined.
104 o
40 23.
○
A
D
In the figure, AB, BC and CD are three equal chords of a circle. If BAC = a, then AED 
E
A.
B.
2a.
3a.
C.
D.
E.
90o  a.
180o  2a.
180o  3a.
D
C
a
A
41 45.
○
A.
D.
B
In the figure, O is the center of the circle. a + b =
a
o
180 .
180o  c.
42 53.
○
B.
c.
E.
c
180o 
.
2
C.
c
.
2
b
c
O
In the figure, C is the centre of the circle. ABCD is a straight line. AQR touches the circle at
Q. If DAR = 20o, then DQR =
A.
B.
C.
D.
E.
35o.
40o.
55o.
65o.
70o.
D
C
B
20o
A
243
Q
R
HKCEE 1988’
43 21.
○
A.
B.
In the figure, O is the centre of the circle of radius 5. AB is a tangent and AO = 12. AC =
A
13.
17.
C.
219 .
D.
E.
244 .
269 .
44 24.
○
12
C
In the figure, TP and TQ are tangents to the circle PQR. If RPQ = 70o and PTQ = 50o,
then RQP =
20 .
45o.
50o.
D.
E.
60o.
70o.
45 51.
○
C.
D.
E.
R
o
A.
B.
C.
A.
B.
B
O 5
Q
70o
50o
P
T
ABCD is a cyclic quadrilateral with AB = AD and CB = CD. Find ABC.
75o
90o
B
A
D
o
105
120o
It cannot be found.
46 52.
○
C
In the figure, O1 and O2 are the centres of the two circles, each of radius r and AB = 12 .
Find r.
A.
1
2
B.
C.
2
4
D.
E.
6
8
O1
A
12
O2
244
B
HKCEE 1989’
47 23.
○
A.
C.
E.
In the figure O is the centre of two concentric circles. ADOEB and CGFB are straight lines.
Which of the following is/are true?
I. AC // DG
II. BF = CG
III. A, E, F and C are concyclic
I only
B. II only
I and II only
D. I and III only
I, II and III
48 24.
○
A.
B.
C.
D.
E.
C
G
F
A
D
B
E
O
In the figure, TC is a tangent to the circle at C and AB // DC. If BCT = 48o, then  =
B
o
48 .
72o.
84o.
48o
90 .
96o.
49 32.
○

A
o
C
 
D
In the figure, ABCD and WXYZ are sectors of equal radii. If BCD : XY Z = s : t, then which
of the following is/are true?
A.
I only
50 51.
○
A.
B.
56o
108o
C.
D.
E.
112o
118o
124o
B
I.
BD s
=
XZ t
II.
area of sector ABCD s
=
area of sector WXYZ t
III.
BAD s
=
XWZ t
B.
II only
X
Y
C
C.
A
III only
D.
D
I and III only E.
W
Z
II and III only
In the figure, O is the centre of the smaller circle. OAB and PQR are straight lines. Find .
B
R

A
O
124 o
Q
P
245
HKCEE 1989’
51 52.
○
A.
B.
C.
D.
E.

In the figure, B is the mid-point of AC . AC = AD. If ADC = 56o, then BCD =
A
o
84 .
90o.
96o.
112o.
124o.
56o
B
D
C
HKCEE 1990’
52 20.
○
 
In the figure, TQ is the tangent to the tangent to the circle at A. If AC  BC and
P
PAQ  48 , then QAC =
o
A.
B.
C.
42 .
48o.
66o.
D.
E.
71o.
84o.
A
T
48o
Q
B
C
53 21.
○
In the figure, O is the centre of the circle. If OR // PQ and ROQ = 42o, find RMQ.
A.
B.
21o
42o
C.
D.
E.
63o
84o
126o
R
O
44o
M
Q
P
54 50.
○
A.
D.
In the figure, PA and PC are tangents to the circle ABC. If P = 48o, then ABC =
A
o
o
o
84 .
B. 96 .
C. 106 .
o
o
114 .
E. 132 .
P
48o
B
C
In the figure, TA and TB are tangents to the circle ABC. If TA  TB and BD  AC, find
55 51.
○
CBD.
A.
C.
E.
o
30
45o
60o
B.
D.
40o
50o
C
B
E
D
A
246
HKCEE 1990’
56 53.
○
A.
C.
E.
3
5
7
In the figure AB, AC and BC are three tangents touching the circle at D, E and F respectively.
A
If AC = 24, BC = 18 and ACB = 90o, find the radius of the circle.
B. 4
D. 6
D
E
HKCEE1991’
57 21.
○
A.
B.
C.
D.
E.
40
70
80
90
140
59 24.
○
A.
B.
C.
D.
E.
C
In the figure, O is the centre of the circle BCD. ABC and EDC are straight lines. BC = DC
and AED  70 . Find BOD .
In the figure, TPA and TQB are tangents to the circle at P and Q respectively. If PQ = PR,
which of the following must be true?
I.
 APR =  QRP
II.  QTP =  QPR
III.  QPR =  APR
I only
II only
III only
I and II only
I and III only
60 52.
○
A.
B.
C.
D.
E.
F
b
2b
180  b
360  b
360  2b
58 22.
○
A.
B.
C.
D.
E.
B
In the figure, O is the centre of the circle. Find a + c.
    
In the figure, A B : B C : CD : D E : E A = 1 : 2 : 3 : 4 : 5 . Find  .
30
36
60
72
120
247
HKCEE 1992’
61 14.
○
A.
B.
C.
D.
E.
In the figure, TP and TQ are tangent to the circle of radius 3cm. Find the length of the minor
arc PQ.
3 cm
2 cm
3
cm
2
 cm

2
cm
62 24.
○
A.
B.
C.
D.
E.
42
36
24
21
18
63 26.
○
A.
B.
C.
D.
E.
In the figure, O is the centre of the circle. Find  .
30
32
35
36
45
64 27.
○
In the figure, the circle is inscribed in a regular pentagon. P, Q and R are points of contact.
Find  .
In the figure, ST is a tangent to the smaller circle. ABC is a straight line. If  TAD 2x and
 DPC 3x, find x.
A.
B.
C.
D.
E.
30
36
40
42
65 50.
○
A.
B.
C.
D.
E.
24
In the figure, the two circles touch each other at C. The diameter AB of the bigger circle is
tangent to the smaller circle at D . If DE bisects  ADC, find  .
38
45
52
66
248
HKCEE 1993’
66 24.
○
A.
B.
C.
D.
E.
20
22.5
25
27.5
30
67 26.
○
A.
B.
C.
D.
E.
In the figure, points A, B, C and D are concyclic. Find x.
In the figure, AB is a diameter. Find  ADC.
100
110
120
135
140
68 50.
○
In the figure, TP and TQ are tangents to the circle at P and Q respectively. If M is a point
on the minor arc PQ and  PMQ =  , then  PTQ =
A.
C.
E.

.
2
180   .
2 180 .
69 51.
○
A.
C.
E.
B.
  90 .
D.
180  2 .
In the figure, O is the centre of the circle. AB touches the circle at N. Which of the following
is / are correct?
I. M, N, K, O are concyclic.
II.  HNB ~  NKB
III.  OAN =  NOB
I only
B. II only
III only
D. I and II only
I, II and III
70 54.
○
In the figure, the three circles touch one another. XY is their common tangent. The two larger
circles are equal. If the radius of the smaller circle is 4 cm, find the radii of the larger circles.
A.
B.
C.
D.
E.
8 cm
10 cm
12 cm
14 cm
16 cm
HKCEE 1994’
71 21.
○
In the figure, O is the centre of the circle. If AC = 3 and  BAC = 30 , find the diameter
AB.
A.
D.
3
2
2 3
B.
6
E.
3 3
C.
3 3
2
30
249
HKCEE 1994’
72 22.
○
A.
B.
C.
D.
E.
In the figure, PA is tangent to the circle at A,  CAP = 28 and BA = BC. Find x.
28
48
56
62
76
73 23.
○
In the figure, O is the centre of the inscribed circle of  ABC. If  OAC = 30 and 
OCA = 25 , find  ABC.
A.
B.
C.
D.
E.
50
55
60
62.5
70
74 51.
○
In the figure, ABCD is a semi-circle, CDE and BAE are straight lines. If  CBD = 30 and
 DEA = 22 , find x.
A. 38
C. 44
E. 60
75 52.
○
A.
B.
C.
D.
E.
B.
D.
41
52
  
In the figure, OABCD is a sector of a circle. If A B = B C = CD , then x =
105 .
120 .
135 .
144  .
150 .
HKCEE 1995’
76 22.
○
A.
B.
C.
D.
E.
25 .
40 .
45 .
50 .
65 .
77 23.
○
A.
B.
C.
D.
E.
In the figure, ABCD is a semicircle.  CAD =
20 .
35 .
45 .
50 .
70 .
In the figure, O is the centre of the circle, POQR is a straight line. TR is the tangent to the
circle at T.  PRT =
250
HKCEE 1995’
78 24.
○
A.
B.
C.
D.
E.
In the figure, ABCD is a cyclic quadrilateral. If  DAB = 110 and BC = BD, find  DAC.
20
35
40
55
70
HKCEE 1996’
79 25.
○
A.
B.
C.
D.
E.
20
27.5
35
37.5
40
80 26.
○
A.
B.
C.
D.
E.
21
24
42
In the figure, O is the centre of the circle. PA is the tangent to the circle at A and CB // PA.
Find x.
45
48
81 50.
○
A.
D.
In the figure, O is the centre of the circle. Find x.
In the figure, O is the centre of the circle. AP, AB and BR are tangents to the circle at P, Q
and R respectively. Which of the following must be true?
I. AP + BR = AB
II. OQ bisects  AOB.
1
III.  AOB =  POR
2
I only
B. II only
C. I and II only
I and III only
E. I, II and III
HKCEE 1997’
82 18.
○
A.
D.
5

2
13 
83 20.
○
A.
B.
C.
D.
E.
In the figure, BCA is a semicircle. If AC = 6 and CB = 4, find the area of the semicircle.
B.
E.
13

2
26 
C.
10 
In the figure, EC is the tangent to the circle at C. Find  CBD.
40
50
65
70
75
251
HKCEE 1997’
84 50.
○
A.
D.
be true?
I.
 BCE ~  ADE
II.  ABC ~  AED
III.  ABC ~  BDA
I only
B. I and II only
II and III only
E. I, II and III
85 51.
○
A.
B.
C.
D.
E.
In the figure, AC is the angle bisector of  BAD. Which of the following statements must


C.
I and III only


In the figure, A B = 2, B C =3, CD = 4 and D A = 6. Find  BCD.
72
84
90
96
144 
HKCEE 1998’
86 28.
○
A.
B.
C.
D.
E.
2 .
4 .
90   .
180   .
180  2 .
87 29.
○
A.
B.
C.
D.
E.
  
In the figure, AD is a diameter of the circle. If A B : B C : CD = 3 : 5 : 7, then  ADC =
36 .
45 .
48 .
49 .
72 .
88 49.
○
A.
B.
C.
D.
E.
In the figure, AB is a diameter of the circle and ABD is a straight line.  CBD =
In the figure, CE is tangent to the circle at C. Find  DCE.
40
42
49
54
78
HKCEE 1999’
89 26.
○
A.
B.
C.
D.
E.
In the figure, O is the centre of the circle. Find x.
12 
20
24
40
60
252
HKCEE 1999’
90 27.
○
A.
B.
C.
D.
E.
26
32
38
52
64
91 50.
○
A.
B.
C.
D.
E.
In the figure, AB is a diameter of the circle. Find x.
In the figure, AT is tangent to the circle at T and ABC is a straight line. Find AT.
9 cm
12 cm
15 cm
16 cm
20 cm
HKCEE 2000’
92 20.
○
A.
B.
C.
D.
E.
40
46
57
66
68
93 31.
○
A.
B.
C.
D.
E.
70
75
90
95
105
95 46.
○
A.
B.
C.
D.
E.
In the figure, CAB is a semicircle and ABCD is a parallelogram. Find the area of ABCD.
65 cm2
60 cm2
52 cm2
32.5 cm2
30 cm2
94 45.
○
A.
B.
C.
D.
E.
In the figure, O is the centre of the circle. EAOB and EDC are straight lines. Find x.
In the figure, AB is tangent to the circle at B. Find  DCE.
  
In the figure, A B : B C : CD = 2 : 1 : 3. Find  ADC.
56
60
63
72
84
253
HKCEE 2001’
96 18.
○
A.
C.
E.
In the figure, AEC is a diameter and DEB is a straight line. Find x.
54 
74 
94 
97 32.
○
A.
B.
C.
D.
E.
70 
92 
B.
D.
In the figure, ABCD is a semicircle, AB : BD = 4: 3. Find AB correct to the nearest 0.1 cm.
5.7 cm
7.6 cm
10.7 cm
13.0 cm
14.3 cm
98 45.
○
In the figure, O is the centre of the circle, AOB is a straight line and BCD is the tangent to
the circle at C. Find x.
A.
C.
E.

99 46.
○
A.
B.
C.
D.
E.
53 
59 
B.
D.
50
56 
62 
 

In the figure, A B = B C = 0.5 CD . Find ABC .
100 
105 
112.5
130 
150 
HKCEE 2002’
1○
00 28. In the figure, O is the centre of the semicircle ABCD and BC // AD. If COD  42  , then
x=
A.
B.
C.
D.

48
63 
84 
90 


1○
01 29. In the figure, AE D  1 and CF D  4 . If ABC  100  , then ABD 
A.
B.
C.
D.
18 
20 
24 
25 
254
HKCEE 2002’
1○
02 51. In the figure, EAF is a common tangent to the circles at the point A. Chords AC and BC of
A.
B.
C.
D.
the smaller circle are produced to meet the larger circle at G and D respectively. Which of
the following must be true?
I.
ADG  EAG
II. ABD  AGD
III. BAE  ADB
I only
II only
I and III only
II and III only
HKCEE 2003’


1○
03 25. In the figure, ABC is a semicircle with BC  7 and ACB  55  . Find A B .
A.
B.
C.
D.
9
10
11
14
1○
04 50. The figure shows a circle with diameter AD. If AB = BC = CD, find x + y + z.
A.
B.
C.
D.
315 
324 
330 
360 
1○
05 51. In the figure, XAB and XDC are straight lines. If DX = 5, AX = 6 and AB = 4, find CD.
A.
B.
C.
D.
5
7
10
3
24
5
1○
06 52. In the figure, BE and BF are tangents to the circle at A and C respectively. If ADC  100  ,
then ABC 
A.
B.
C.
D.

20 .
30  .
40  .
50  .
255
HKCEE 2004’
1○
07 23. In the figure, O is the centre of the circle ABCD. If EAB and EDOC are straight lines and
EA = AO, find AEO .
A.
B.
C.
D.

18
24 
27 
36 
1○
08 24. In the figure, O is the centre of the circle ABC. Find x.
A.
B.
C.
D.
17.5
27.5 
35 
55 
1○
09 25. In the figure, ABCD is a circle. AC and BD meet at E. If AD = 4, AE = 2, EC = 5 and BE  4 ,
then BC =
A.
B.
C.
D.
6.
7.
8.
10.

1○
10 26. In the figure, ABC is a circle. If ABC  30  and AC  4 , then the circumference of the
circle is
A.
B.
C.
D.
24.
48.
8 .
16 .




1○
11 50. In the figure, ABCD is a circle. If CD  2 DA  2 AB  2 BC , then x =
A.
B.
C.
D.
108  .
112  .
120  .
144  .
1○
12 51. In the figure, TS, SQ and QP are tangents to the circle at T, R and P respectively. If TS // PQ ,
TS = 3 and QP = 12, then the radius of the circle is
A.
B.
C.
D.
4.5.
6.
7.5.
9.
256
HKCEE 2005’
1○
13 24. In the figure, ABCD is a circle. AB produced and DC produced meet at E. If AC and BD
intersect at F, then ABD 
A.
B.
C.
D.

41 .
52  .
56  .
60  .
1○
14 25. In the figure, ABCD is a circle. If AC is a diameter of the circle and AB = BD, then CAD 
A.
B.
C.
D.
18  .
21 .
27  .
36  .
1○
15 49. In the figure, AB and AC are tangents to the circle at X and Y respectively. Z is a point lying
on the circle. If BAC  100  , then XZY 
A.
B.
C.
D.
40  .
45  .
50  .
55  .
1○
16 50. In the figure, O is the centre of the circle and AOC is a straight line. If AB and BC are
tangents to the circle such that AB = 3 and BC = 4, then the radius of the circle is
A.
B.
C.
D.
3
.
2
12
.
7
2.
5
.
2
   
1○
17 51. In the figure, ABCD is a circle. If A B : BC : CD : D A = 1: 2: 3: 3 and E is a point lying on
BD, then CAE 
A.
B.
C.
D.

45 .
50  .
55  .
60  .
257
HKCEE 2006’
1○
18 46. In the figure, O is the centre of the circle ABC. If OBC  50  and ACO  20  , then
BOA 
A.
B.
C.
D.

50 .
60  .
70  .
80  .
1○
19 47. In the figure, O is the centre of the circle. A and B are points lying on the circle. If AOC is
a straight line and BC is a tangent to the circle, then the radius of the circle is
A.
B.
C.
D.
3
.
2
3.
2 3
3 3.
HKCEE 2007’
1○
20 49. In the figure, A, B and C are points lying on the circle. AB is a diameter of the circle. DB is
A.
B.
C.
D.
2
4
4
8
the tangent to the circle at B. If ACD is a straight line with AC = 4 and CD = 2, then AB 
6.
3.
6.
3.
HKCEE 2008’
1○
21 50. In the figure, O is the centre of the circle ABCD. If ADC  84  and CBO  38  , then
AOB 
A.
B.
C.
D.
64 
88 
104 
168 
1○
22 51. In the figure, AB is the tangent to the circle at B and ADC is a straight line. If AB : AD  2 : 1,
then the area of ABD : the area of BCD =
A.
B.
C.
D.
1 : 2.
1 : 3.
1 : 4.
2 : 3.
258
HKCEE 2009’
1○
23 48. In the figure, AB is a diameter of the circle ABCD. It is given that AC and BD intersect at E.
If AED   , then
A.
B.
C.
D.
sin  .
cos .
tan  .
1
.
tan 
CD

AB
1○
24 49. In the figure, ABCD is a circle. If AB = AC, AB // DC and ABD  40  , then CBD 
A.
B.
C.
D.
10  .
20  .
30  .
40  .
1○
25 50. In the figure, AB is the tangent to the circle at A. If AB = 20 and BC = 50, find the radius of
the circle.
A.
B.
C.
D.
20
25
29
30
HKCEE 2010’
1○
26 49. In the figure, AD is a diameter of the circle ABCD. It is given that XBCY is a straight line.
A.
B.
C.
D.
If AD = 20cm and BC = 12cm, then AX + DY =
12 cm.
16 cm.
32 cm.
36 cm.
1○
27 50. In the figure, XY and XZ are the tangents to the circle ABCD at A and B respectively. If
AXB  50 and DAY  30 , then BCD 
A.
B.
C.
D.
65.
80.
95.
130.
259
HKCEE 2011’
1○
28 48. In the figure, ABCDE is a circle. AC and BD intersect at F. If AE//BD, DAE  20 and
A.
B.
C.
D.
20 .
35 .
45 .
50 .
CFD  70 , then CBD 
1○
29 49. In the figure, BC is a diameter of the circle ABC. BCD is a straight line and DA is the tangent
to the circle at A. If ABC  28 , then ADB 
A.
B.
C.
D.
22.
28.
34.
62.
HKDSE 2012’
1○
30 20. In the figure, O is the centre of the circle ABCD. If BAO  28 , BCD  114 and
CDO  42 , then ABC 
A.
B.
C.
D.
90 .
96 .
100 .
138 .
1○
31 41. In the figure, PQ is the tangent to the circle ABC at O, where O is the centre of the semicircle
PBQ. It is given that BCP is a straight line. If BPQ  12 , then ABC 
A.
B.
C.
D.
18 .
24 .
36 .
54 .
HKDSE 2013’
1○
32 19. In the figure, ABCD is a circle. AC and BD intersect at E. If AB = AD and AD // BC , then
BAE 
A.
B.
C.
D.
53 .
57 .
69 .
74 .
260
1○
33 41. In the figure, O is the centre of the circle ABC. DE is the tangent to the circle at A. If AB is
the angle bisector of CAE , then ACO 
A.
B.
C.
D.
26 .
28 .
31 .
34 .
HKDSE 2014’
1○
34 20. In the figure, AC is the diameter of the circle ABCDE. If ADE  28 , then CBE 
A.
B.
C.
D.
56 .
62 .
72 .
76 .
1○
35 21. In the figure, O is the centre of the circle ABCDEF. PQR intersect the circle at A, B, C,
A.
B.
C.
D.
D, E and F. If QPR  38 and AB = CD = EF, then QOR 
109 .
117 .
123 .
142  .
1○
36 41. In the figure, PQS is a circle. PQ is produced to R such that RS is the tangent to the circle
at S. I is the in-centre of QRS . If IRQ  12 and PSQ  70 , then QPS 
A.
B.
C.
D.
24 .
37 .
43 .
62 .
HKDSE 2015’
1○
37 20. In the figure, AD is a diameter of the circle ABCDE. If BAD  58 and BC = CD, then
A.
B.
C.
D.
AEC 
32 .
58 .
61 .
64 .
261
HKDSE 2015’
1○
38 40. In the figure, AB and AC are the tangents to the circle at B and C respectively. BD is a
A.
B.
C.
D.
diameter of the circle. AC produced and BD produced meet at E. If AB = 6 cm and
= 10 cm, then BD =
3 cm.
5 cm.
6 cm.
8 cm.
262
AE
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263