SYMMETRY OF CIRCLES.ppt
17
CHAPTER 3 CIRCLES II
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Transcript of SYMMETRY OF CIRCLES.ppt
CHAPTER 3
CIRCLES II
Symmetry Of Circles
A circle is symmetrical about any line that passes through its centre.Example:
Axes of symmetry
Any diameter of a circle is an axis of symmetry
Example:Identify the axis of symmetry for the following circles.
L
M
KJ
OS
R
QP
U
T
O
LMTU
Properties Of Chords
A radius that is perpendicular to a chord divides the chord into two equal parts and vice versa.
P
T BA
O
When OP AB, then AT = TB. When AT = TB, then OP AB.
Example:
y cm
8 cm
x cm10 cm
O
Find the lengths of x and y.
x = ______________________
y = ______________________
8 cm
6 cm
The perpendicular bisectors of two chords intersects at the centre of the circle.
O
FD
E
B
C
A
AC and DF are two chords. BO and EO meet at the centre O.
Two chords that are equal in length are equidistant from the centre and vice versa.
Q
XP
Y SR
0
When PQ = RS, then OX = OY. When OX = OY, then PQ = RS.
Example:
X
O
H
Y
G
F
E
Given that EF = GH = 12 cm and OF = 10 cm, find the length of OX.
OX = _________8 cm
Chords of the same length cut arcs of the same length.
F E
D
CB
A
When chord AB = chord CD = chord EF, then minor arc AB = minor arc CD = minor arc EF.
Example:
O
N M
L
KJ
Given that JL = NL and NMLis 18 cm, find the length ofJKL.
JKL = _________18 cm
XPY is called a minor arc whileXQY is called a major arc.
majorsegment
minorsegment
Q
O
P
Y
X
Example:1.In the diagram, AB = DC. AMB, DNC and DOB
are straight lines. If O is the centre of the circle, MN = 16cm and DB = 20cm, find the length of AB.
N
M
O
D C
BA
OB = 1
2 DB = 10 cm
OM = 1
2 MN = 8 cm
MB = 102 - 82
= 100 - 64
= 36
= 6 cm
AB = 2 x 6 = 12 cm
2.In the diagram, P and Q are the centres of two identical circles of radius 15cm. If KL = MN = 24cm, find the length of XY.
N
M
YQPX
L
K
KX = 24
2 = 12 cm
XP = 152 - 122
= 225 - 144
= 81
= 9 cm
XY = XP + PQ + QY = 9 cm + 15 cm + 9 cm
= 33 cm
3.Given that PQ = 12 cm and RS = 8 cm, find the length of PM and RN.
M
Q
P
N SR
O
PM = 12 cm
2
= 6 cm
RN = 8 cm
2
= 4 cm
4.In the figure above, BD = 4 cm. Find the length of AOB.
D
C
E
4 cm
5 cm
O BA
EB = BD = 4 cm
OB = 3 cm
AOB = 5 cm + 3 cm = 8 cm
5.Given that the radius of the circle is 10 cm and RT is 12 cm, find the length of PS.
S
Q
R
O
TP
RS = 6 cm
OS = 8 cm
PS = 10 cm + 8 cm = 18 cm
5.In the figure, the arcs SVT and PQR are of the same length. Given that OS = 13 cm, find the length of UR and OU.
PR = 10 cm
UR = 5 cmU
10 cm
Q
T
V
SR
P
O OU2 = OR2 - UR2
= 132 - 52 = 169 - 25OU = 144 = 12 cm
