View
232
Download
0
Category
Preview:
Citation preview
7/28/2019 Chapter 14 I Circles ENHANCE
1/20
220
x
O
CHAPTER 11: CIRCLES III
11.1 PROPERTIES OF ANGLES IN CIRCLES
1. Angle subtended by an arc at the centre is twice the angle subtended at the circumference
by the same arc.
x = 2y
Example
a) Find the value of x.
x = 110 2 = 55
Exercises
i). Find the value of x.
ii)
iii)
2. Angle subtended at the circumference in a semicircle is 90 .
x = y = 90
Example
b) Find the value of x.
x = 90 60
= 30
Exercises
Find the value of x.
i)
x =
Circles III
O
35x
42
O
x
1
x =
x =
ox 54
x
O
110
x =
y
x
O
xy
60
x
O
7/28/2019 Chapter 14 I Circles ENHANCE
2/20
ii)
x =
ii)
x =
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
Example
Find the value of
y = 47o
Exercises
i) Find the value of xand y.
x = y =
Circles III
x 47
y
38
2
47o y
35 xo x
o
rs
x
y
7/28/2019 Chapter 14 I Circles ENHANCE
3/20
ii) Find the value of y.
y =
iii) Find the value of x and y.
x y+ =
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
Example
Find the value of xand y.
x = 180 - 85 y = 180 - 95
= 95 = 85
Exercises
Find the value of x and y.
i)
x = y =
ii)
iii)
Circles III
27y
x
80y
115
A
x
B O50 110
yD
C
3
P Q95 x
y
85 R
S
s
p
q r
60o
17oxy
x
y
7/28/2019 Chapter 14 I Circles ENHANCE
4/20
x =
y =
x =
y =
11.2. PROPERTIES OF TANGENTS TO CIRCLES.
11.2.1 Tangent to a circle
11.2.2
i) Tangents drawn from an external point are equal, AB = CB
ii) The tangents subtend equal angles at the centre, 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
Example
Find the value of x.
Exercise
a i)
Circles III
A
x
B O50 110
yD
C
P
xS y 100
110R Q
O
4
O
Tangent
A tangent to a circle is
perpendicular to the
drawn to the point of
contact.
B
C
A
P
O pqo
7/28/2019 Chapter 14 I Circles ENHANCE
5/20
x = 90 - 65
= 25
x =
Example
x = 180 - 50
= 130
Exercises
b i) Find the value of x.
x =
ii) Find the value of x and y
x = y =
iii) Find the value of x and y.
x = y =
11.2.3 ANGLES BETWEEN TANGENTS AND CHORDS
Circles III
65
x
O
P
O Q
70 x
R
P
O Qx
50
P
O Qx
15
R
y54
x
Q
R
O
13y
x
N
O
P
5
P
7/28/2019 Chapter 14 I Circles ENHANCE
6/20
47
x
35
y
P
Q
R
Example 1
Given STU is a tangent to the circle with centre O.
Find the value of x and y.
x = 47 y = 63
1a) Given PQR is a tangent to the circle with
centre O. Find the value of x and y.
x = y =
Example 2
x = 58 y = 180- 116
= 64
2a)
x = y =
Circles III
a
by
x
S
T
U
63
47x
y
x y
75
60
P Q R
y
x58
T
U
6
S
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
7/28/2019 Chapter 14 I Circles ENHANCE
7/20
x48
Q
O
N
M
R
P
b)
x =
y =
c)
x =
y =
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.
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
Circles III 7
88o
15o
x yx
y81o
31o
DIAGRAM 1
DIAGRAM 2
38
x
N
Q
P
R
M
7/28/2019 Chapter 14 I Circles ENHANCE
8/20
P
R45
110
S
xO
Q
48
xO
RP
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.
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 8
A CB
E
65O
56O
D
F
xO
DIAGRAM 3
DIAGRAM 4
Q
DIAGRAM 5
7/28/2019 Chapter 14 I Circles ENHANCE
9/20
R
Q
P
65
y
x
38
D
80 36
yEF
BA
x
N
MLO
38
K
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 9
DIAGRAM 7
DIAGRAM 8
7/28/2019 Chapter 14 I Circles ENHANCE
10/20
70y
45
O
P
Q
SRT
x
5342
y
D
C
B
E
A
y
100
70
xQ
P
T
S
R
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
10. 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 10
DIAGRAM 11
7/28/2019 Chapter 14 I Circles ENHANCE
11/20
D
C
O
A
y
42
T
SP
y
O
64
RQ
x
QP
O
R
148
B
A
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 11
DIAGRAM 12
DIAGRAM 13
DIAGRAM 14
7/28/2019 Chapter 14 I Circles ENHANCE
12/20
P
x
30O
58
NMK
x
24
50
OP
N
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 12
KL
M
P
K
Q
R35o
65o xo
DIAGRAM 15
DIAGRAM 16
DIAGRAM 17
Q
7/28/2019 Chapter 14 I Circles ENHANCE
13/20
40
x
S
R
Q
B
P
A
xO
R
40
QP
S
6548x
T
R
S
QP
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 13
DIAGRAM 18
DIAGRAM 20
DIAGRAM 19
7/28/2019 Chapter 14 I Circles ENHANCE
14/20
T
S
55O
100
yR
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
June 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 14
J
L
K
M
O
15
55
n
7/28/2019 Chapter 14 I Circles ENHANCE
15/20
Q
Ox
40
DIAGRAM 2
The value of x is
A 15 C 60
B 45 D 75
June 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 15
J
M
K
L
30 x
7/28/2019 Chapter 14 I Circles ENHANCE
16/20
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
June 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 16
50
Q
O
S
P
T
T
P
S
68
Q
x
R
7/28/2019 Chapter 14 I Circles ENHANCE
17/20
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
B 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 126
November 2007
Circles III 17
25
X
U
P
O
R
Q
S
T
66o
27o
xo
T
U
QO
S
R
P
Diagram 3
7/28/2019 Chapter 14 I Circles ENHANCE
18/20
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 18
16
O
FGE
58x
QP
Diagram 3
RTU
20
xQP
S
Diagram 4
7/28/2019 Chapter 14 I Circles ENHANCE
19/20
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
ANSWERS
Circles III 19
10O
Q UT
43
x
P
32S
Diagram 5
50
O
K
LJ35
xN
M
Diagram 5
7/28/2019 Chapter 14 I Circles ENHANCE
20/20
Exercise 11.1
1 i) 70 ii) 110 iii) 84
2. i) 36 ii) 55 iii) 30
3 i) x = 47 y = 38
ii) y = 27
iii) x = 17 y = 60
4 i) x = 115 y = 100
ii) x = 55 y = 125
iii) x = 30 y = 50
Exercise 11.2
a i) x = 20
b) i) x = 150
ii) x = 90 y = 126
iii) x = 154 y = 13
Exercise 11.2.3
1 a) x = 60 y = 45o
2 a) x = 47 y = 35
b) x = 61 y = 44
c) x = 99 y = 68
1. C 11. A
2. C 12. C
3. 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
2003 NOV B2004 JUNE A 2004 NOV B
2005 JUNE B 2005 NOV C
2006 JUNE C 2006 NOV A
2007 JUNE B 2007 NOV B
2008 JUNE C 2008 NOV A
Circles III 20
Recommended