PROJECT OSCAR

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PROJECT OSCAR
Kendriya Vidyalaya Bhandup
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MATHEMATICS
CIRCLES
Hi! I am a circle
CLASS- VIII
3
Slide no.
Topic
4
What is a circle???
5
About a circle
6
Some types of circles
7
Parts of a circle
8
Some more parts of circle
9
Some more parts of circle
10
Some more parts of circle
11
Cyclic Quadrilateral
CONTENTS
12
Properties of cyclic quadrilateral
13
Property 2
14
Property 3
15
Circumference of a circle
16
Property 2
17-20
Examples
21-22
Test your brains
23
Done by
24
The End
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WHAT IS A CIRCLE???
Have you ever seen a wheel of bullock cart?? It
is a circle. So, here is -
MY DEFINITION The locus of a moving point which
moves in a plane so that its distance from a
fixed point is called circle and thats me.
r
c
5
ABOUT A CIRCLE

• The fixed point is called centre.
• The constant distance is called the radius.
• Let r be the radius then circumference
  222/7r.
• Let r be the radius then area 22/7rr.
• If c is the centre of a circle of radius r
  and P is a point in its plane such that
  
• OPltr, P is in the interior of the circle.
  
• Opr, P is on the circle.
  
• OPgtr, P is in the exterior of the circle.

r
c
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SOME TYPES OF CIRCLES
CONGRUENT CIRCLES Circles having equal radii are
called congruent circles.
a
a
P
Q

• SEMI CIRCLE Each diameter divides the circle
  into segments.
• Circumference 22/7r2r.
• Area ½22/7rr
• Angle of semi circle is a right angle.

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PARTS OF CIRCLE
RADIUS A line segment joining the centre of
the circle and any point on the circle is known
as the radius of the circle.
Radius
CHORD The line segment joining any two points of
the circle is called a chord.
Chord
DIAMETER A chord passing through the centre of
the circle to any point on the circle is known as
the diameter of the circle.
Diameter
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SOME MORE PARTS OF A CIRCLE
1.ARC It is the part of the circle between any
two given chords. The smaller arc is called as
minor arc and bigger arc is called major arc.
Major arc
Minor arc
2.SECTOR It is a region enclosed by two radii
and arc of the circle.
Sector
3.SEGMENT It is a region enclosed by a chord and
the arc joining the chord. The segment made my
minor arc is called minor segment and segment by
major arc as major segment.
Major segment
Minor segment
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4.CIRCUMFERENCE OF A CIRCLE The length of the
circle is called the circumference of the
circle. 5.POINT OF TANGENCY The point which is
common to the line and the circle is called the
point of tangency.
O
P
TANGENT
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6.SECANT A line which has two points in common
with the circle is called its secant. 7.CENTRAL
ANGLE The angle subtended by an arc at the
centre of the circle is called the central
angle.
A
O
B
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CYCLIC QUADRILATERAL
CYCLIC QUADRILATERAL The quadrilateral whose all
the four vertices lie on the circle is called a
cyclic quadrilateral.
A
D
B
C
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PROPERTIES OF A CYCLIC QUADRILATERAL
PROPERTY 1
1. Opposite angles of the cyclic quadrilateral
are supplementary.
D
60
C 120
D 60
A
120
B 120
A 60
Hence
Hence
B
C
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PROPERTY 2
2. An exterior angle of a cyclic quadrilateral is
congruent to the angle opposite to its adjacent
interior angle.
A
D
B
C
E
DCE
is congruent to
A
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PROPERTY 3
3. The central angle subtended by an arc is
double the angle subtended by it on the remaining
part of the circle.
Q
O 120
O
Hence
Q 60 Because Q1/2of O
120
R
P
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CIRCUMFERENCE OF A CIRCLE
PROPERTY 1
1. The perpendicular drawn from the centre of the
circle to a chord bisects the chord.
O
AM BM half of AB.
A
M
B
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PROPERTY 2
2. Congruent chords of the same circle
are equidistant from the centre.
P
B
AB PQ
.
N
Hence OM ON
M
O
Q
A
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Examples
1.The radius of a circle with centre O is 6cm.
Point P lies on the plane of the circle. If OP
4.5 cm, find the position of P with respect to
the circle. Sol OP 4.5 cm. Radius 6cm.
Thus OP is lesser than the radius. Hence P lies
in the interior of the circle.
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2. The length of a chord of a circle with radius
17 cm is 30 cm. Find its distance from the
centre.
o
17
17
30

• Sol In the above figure we see that OA 17 cm,
  AM ½ of AB 15 cm.
• OM 17 15
• 64 8 Thus OM 8 cm

2
2
2
2
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3. The measure of the angle between chord AB and
radius OA of a circle is 30. Find its distance
from the centre.
o
oo
0
8
30
A
B
M
o
o
SOL
M 90
o
A of OAM 30
o
o
o
o
Hence,
O 30 90 180 60.
0000000000
Thus, OM 1/2 of OA 1/2x8 4cms.
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4.In the adjoining figure angle BAD 35 .Find
angle BCD.
B
A
35
C
D
Sol Since ABCD is a cyclic quadrilateral, BCD
BAD 180 Hence BCD 180 35 145
0
0
0
0
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TEST YOUR BRAIN
1. In the figure below arc BXC and arc BYC are
opposite arcs. If arc BXC is congruent to arc BYC
find the measure of angle BAC.
A.
OPTIONS
.X
1. 60 2. 90
.
.C
B.
O
3. 45 4. 30
Y.
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2. In the figure below chord AB is 3 cm away from
the centre of the circle and radius is of 5 cm.
find the length of the chord.
OPTIONS

• 16 cm 2. 8 cm
• 3. 2 cm 4. 4 cm

O
3
5
B
A
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A Team Work By-
PRASANNA
HIMAJA
PRIYANKA
RAMYA
of K.V.BHANDUP
SARVANI
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THE END