chapter 14 i circles enrich
TRANSCRIPT
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x= 2y
o
x
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CHAPTER 14: CIRCLES III
14.1 PROPERTIES OF ANGLES IN CIRCLES
1. Angle subtended by an arc at the centre is twice the angle subtended at the circumferenceby the same arc.
1b
x = 2y
2. Angle subtended at the circumference in a semicircle is 90o.
x = y = 90o
3. a) Angles subtended at the circumference by the same arc are equal. b
r = s
b) Angles subtended at the circumference by arcs of the same length are the same.
x = y
Circles III 1
y
x
O
xy
O
r s
x
y
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4 a) The sum of interior opposite angles of a cyclic quadrilateral is 180o.
p + r = 180o
q + s = 180o
b) The exterior angle and the corresponding interior opposite angle of cyclic quadrilateral are
equal
x = y
5. Angle between a tangent and a chord through the point of contact is equal to the angle
in the alternate segment.
a = b
x = y
6. A tangent to a circle is perpendicular to the radius drawn to the point of contact.
7.
i) Tangents drawn from an external point are
equal, AB = CB
ii) The tangents subtend equal angles at thecentre, AOB = COB
iii) The line joining the external point to the
centre of the circle bisects the angle between
the tangents,
ABO = CBO
p + q = 180
Questions based on examination format (Paper 1)
1. In Diagram 1, the tangent PQR touches the circle at Q and MNR is a straight line. Find the
value of x.Circles III 2
s
p
q r x
y
a
b
x
y
B
C
A
P
Op
qo
38
x
N
Q
P
R
M
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R45
110
S
xO
Q
48
xO
RP
x48
Q
O
N
M
R
P
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A 38 C 109
B 52 D 142
2. In diagram 2, PQR is the tangent to the circle with centre O. Given that the length of arc MN is
equal to length of arc NQ. Find the value of x.
A 21 C 69B 48 D 100
3. In diagram 3, O is the centre of a circle and PQR is a tangent . Find the value of x.
A 45 C 115
B 90 D 220
4. In Diagram 4, PQR is the tangent to the circle with centre O.
Find the value of x.
Circles III 3
DIAGRAM 1
DIAGRAM 2
DIAGRAM 3
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A 21 C 42
B 24 D 60
5. In Diagram 5, ABC is the tangent to the circle BDEF. The value of x is
A 48 C 56
B 54 D 59
Circles III 4
A CB
E
65O
56O
D
F
xO
DIAGRAM 4
Q
DIAGRAM 5
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R
Q
P
65
y
x
38
D
80 36
yEF
BA
x
N
MLO
38
K
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6. In Diagram 6, KOLM is a straight line and MN is the tangent to the circle with centre
O. Find the value of x..
A 14 C 38
B 19 D 142
7. In diagram 7, PQR is the tangent to the circle .Find the value of x + y.
A 103 C 141
B 115 D 168
8. In diagram 8, ABC is the tangent to the circle at B. Given that EF = ED and CDF is
the straight line. Find the value of y .
A 28 C 60
B 44 D 62
Circles III 5
DIAGRAM 7
DIAGRAM 8
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70y
45
O
P
Q
SR
x
5342
y
D
C
B
E
A
y
100
70
xQ
P
T
S
R
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9. In the diagram, TRS is the tangent to the circle with centre O at R.
The value of y is
A 80 C 110
B 100 D 115
In diagram 10, tangent AE touches the circle ABCD at A. Given BC is parallel to AD.
Find the value of x + y.
A 36 C 53
B 42 D 95
11. In Diagram 11, PT is the tangent to the circle PQRS and RST is the straight line.
The value of x + y is
A 110 C 150
B 130 D 170
Circles III 6
DIAGRAM 11
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D
C
O
y
42
T
SP
y
O
64
RQ
x
QP
O
R
148
B
A
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12. In Diagram 12, tangent CD touches circle ABC with centre O at C.
The value of y is
A 25 C 48
B 32 D 53
13. In Diagram 13, PST is tangent to the circle with centre O, at S and POR is straight line.
The value of y is
A 28 C 46
B 38 D 62
14. In the diagram below, ABC and APQ are tangents to the circle with centre O.
The value of x is
A 48 C 74
B 58 D 84
Circles III 7
DIAGRAM 12
DIAGRAM 13
DIAGRAM 14
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Q
P
x
30O
58
NMK
x
24
50
OP
N
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15. Diagram 15 shows the tangent KMN to the circle with centre O at M.
The value of x is
A 52 C 58
B 56 D 62
16. In Diagram 16, KLM is a tangent to the circle. The value of x is
A 76 C 80
B 78 D 82
17. In Diagram 17, MNP is a tangent to the circle QRN at N with centre O. The value of x
is
A 40 C 74
B 64 D 124
18. In Diagram 18, APB is a tangent to the circle QRN at P. Given QPS = SPB.
Circles III 8
KL
M
P
K
Q
R35o
65o xo
DIAGRAM 15
DIAGRAM 16
DIAGRAM 17
Q
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40
x
S
R
Q
B
P
A
xO
R
40
QP
S
6548x
T
R
S
QP
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Find the value of x.
A 70 C 140
B 110 D 160
19. In Diagram 19, PQ is a tangent to the circle PRS with centre O. SRQ is a straight line.
Find the value of x.
A 25 C 50
B 40 D 100
20. In diagram 20, PQR is a tangent to the circle QST. Given that PTS is a straight line.
Find the value of x.
A 17 C 67
B 42 D 113
Past Year SPM Questions
Circles III 9
DIAGRAM 18
DIAGRAM 20
DIAGRAM 19
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T
S
55O
100
yR
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November 2003
In Diagram 3, RST is a tangent to the circle with centre O, at point S.
Diagram 3
The value of y is
A 45 C 70
B 60 D 125
July 2004
In the Diagram 3, O is the centre of the circle and JKL is a tangent to the circle at point K.
DIAGRAM 3
The value of n is
A 25 C 50
B 40 D 65
November 2004
In Diagram 2, JK is a tangent to the circle KLM at K and JML is a straight line
Circles III 10
J
L
K
M
O
15
55
n
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Q
Ox
40
P
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DIAGRAM 2
The value of x is
A 15 C 60
B 45 D 75
July 2005
In Diagram 2, O is the centre of circle QST and PQR is the tangent to the circle at Q.
DIAGRAM 2
Find the value of x.
A 20 C 50
B 25 D 80
Circles III 11
J
M
K
L
30 x
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November 2005
In Diagram 3, TPS is a tangent to the circle PQR at P.
DIAGRAM 3
The length of arc PR is equal to the length of arc QR. Find the value of x.
A. 22 C. 44
B. 34 D. 68
July 2006
In Diagram 2, SPT is a tangent to the circle with centre O, at P.
DIAGRAM 2
Find POQ
A 60 C 100
B 80 D 140
November 2006
Circles III 12
50
Q
O
S
P
T
T
P
S
68
Q
x
R
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In Diagram 2, QRT is a tangent to the circle centre O, at R. PUS, OPQ and OUR are straight lines.
DIAGRAM 2
Find the value of x.
A 40 C 50
45 D 65
June 2007
In Diagram 3, TQU is a tangent to the circle centre O, at point Q. SPQ is straight line.
Find the value of x.
A 114 C 120
B 117 D 126November 2007
Circles III 13
25
X
U
P
O
R
Q
S
T
66o
27o
xo
T
U
QO
S
R
P
Diagram 3
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Diagram 3 shows a circle, PFQ, centre O. EFG is a tangent to the circle at F.
Find the value of x.
A 32 C 61
B 48 D 74
June 2008
1. In Diagram 4, URT is the tangent to the circle PQRS at point R. PQR is an equilateral triangle and
URS = 20o .
Find the value of x.
A 30 C 40
B 35 D 45
2. In Diagram 5, O is the centre of a circle. TQU is the tangent to the circle at point Q. PSR is a straight
line and OPS = 10o .
Circles III 14
16
O
FGE
58x
QP
Diagram 3
RTU
20
xQP
S
Diagram 4
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Find the value of x.
A 18 C 28
B 22 D 33
November 2008
In Diagram 3, JKL is the tangent to the circle centre O at K and JNM is a straight line .
Find the value of x.
A 45 C 15
B 25 D 5
Circles III 15
10O
Q UT
43
x
P
32
S
Diagram 5
50
O
KL
J
35
xN
M
Diagram 5
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ANSWERS
EXAMINATION FORMAT QUESTIONS.
1. C 11. A
2. C 12. C3. C 13. B
4. A 14. C
5. D 15. D
6. A 16. C
7. C 17. B
8. A 18. B
9. B 19. A
10. D 20. A
PAST YEARS SPM QUESTIONS.
2003 NOV B
2004 JUL A 2004 NOV B
2005 JUL B 2005 NOV C
2006 JUL C 2006 NOV A
2007 JUN B 2007 NOV B
2008 JUN C 2008 NOV A
Circles III 16