chapter 8 circles iii
TRANSCRIPT
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CHAPTER 8 : CIRCLES III8.1 TANGENTS TO A CIRCLE
8.2 ANGLES BETWEEN TANGENTS
AND CHORDS
8.3 COMMON TANGENTS
Created By: Mohd Said B Tegoh
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Tangent
A B
O
A tangent to a circle is a straight line which
touches the circle at a point. The point is
called the point of tangency or the contact
point.
8.1 (a) Identifying Tangents to A Circle
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Tangent
A B
O
The tangent to a circle is a straight line
perpendicular to the radius that passes
through the contact point.
8.1(a) Identifying Tangents to A Circle
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A
P
Q
O
The lengths of the tangents from a givenpoint A to each contact points are equal,
AP = AQ
8.1(b) Properties of Tangents to A Circle
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A
P
Q
O
The line that joins the given point outside
the circle, point A, to the centre of the circlebisects;
The angle between the two tangents, that is,
< PAO = < QAO
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A
P
Q
O
OPA and OQA are congruent
The angle between the two radii that
passes through the contact points, that is
< POA = < QOA
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(a) A cord is a line that connects two distinctpoints on a circle. The cord CD divides the
circle into two portions
(b) The smaller portion is called the minor
segment whereas the larger portion is called
the major segment
C DMinor segment
Major segment
8.2 Angles Between Tangents and Chords
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A
B
H
The segment FHG (the major segment) is called the
alternate segment for < AGF. The angle in the
alternate segment which is subtended by the chord
FG is <FHG
G
F
8.2 (a) Identifying the angle in the alternate
segment
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D E
A
a
b
p
q a = b
P = q
C
B
The angle between a tangent and a chord
through the point of contact is equal to
the angle subtended by the chord
alternate segment
8.2 (b) Calculations involving the angle in
the alternate segment
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a
b
c
da = b
c = d
A B
C
8.2 (b) Calculations involving the angle in
the alternate segment
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A Common Tangent to two circles is a
straight line that touches each of the
circles at only one point.
Common Tangent
8.3 COMMON TANGENTS
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P
Q
RS
O C
T Two Common
Tangents
PQ = RS
TOC is a straight line
No. of common tangents
= 2.
(a) Two Circles which intersect
at Two Points
8.3 (b) Properties of Common Tangents
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P Q
R S
O C
Two parallel common
tangents
PQ = RS
PQ, OC, and RS are parallels
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P Q
R S
O C
Three Common
Tangents
PQ = RSXP = XQ = XY = YZ = ZR = ZS
(b) Two Circles which intersect at one point
X
Z
Y
OYC is a straight lineNo. of common tangents = 3, that are PQ, RS,
and XZ
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P
Q
R
S
OC
T Three Common Tangents
PQ = RS
XP=XQ=XY=YZ=ZR=ZSNo. of Common Tangents = 3, that are PQ,RS, and XZ
TOYC is a straight line
X
Y
Z
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(c) Two Circles which Do Not Intersect
P
Q
R
S
OC
T Four common
tangents
PQ = RSKL = MN
No. of common tangents = 4, that are PQ,RS, KL and MN
OXC is a straight line
K
N
ML
X
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P Q
R S
O C
PQ = RSKL = MNOXC is a straight line
No. of common tangents = 4, that are PQ, RS, KL and MN
K
L
M
N
X
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Q
R
T
S
x 0
DIAGRAM 3
In Diagram 3, P QR is a tangent to the circle at point Q.
Given that P QT = 56º. The length of the arc TQ is
equal to the length of the arc SQ.
P
C l o n e d S
P M
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Q
R
T
S
x 0
DIAGRAM 3
P
560
560
56
0
x = 180 ² 56 - 56
= 68
Solution
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DIAGRAM 4
In Diagram 4, LJK is a tangent to the circle JNM at J
and KMN is a straight line. Arc MN = Arc MJ
The value of x isA 15º
B 25º
C 35º
D 65º
K
M
N
x
55º
J
L
C l o n e d
S P M