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Class – IX
CIRCLE
It is a collection of points whose distance from a fixed point always remains constant.
The fixed point is called the center of the circle and the fixed distance is called the radius of
the circle. The length of the complete circle is called its circumference.
Elements of a circle
Let P and Q are two points on the circumference of the circle.
1.
Chord – Line Segment (PQ) joining any two points on a
circle is called chord.
Chord that passes through the center of the circle is called
the diameter.
 Diameter is the longest chord.
 All diameters have the same length, which is equal to
two times the radius.
Major arc
PQ
Chord
Q
P
Minor arc
PQ
2.
Arc – A piece of a circle between two points is called Arc. The longer one is
called the major arc and the shorter one is called the minor arc.
 When P and Q are ends of a diameter then both arcs are equal and each is called a
semicircle.
3.
Segment - The region between a chord and either of its arcs is called
a segment of the circle. The larger region is called as major
segment and the smaller region is called as minor segment.
Major
segment
P
5.
Sector: The region between an arc and the two radii, joining the
center to the end points of the arc is called a sector. Larger one is
called as major sector and the smaller one as minor sector.
 When two arcs are equal, that is, each is a semicircle, then
both segments and both sectors become the same and each is
known as a semicircular region.
Tangent - A line which intersects the circle at exactly one
point is called a tangent.
 At a point of a circle there is one and only one tangent.
 From an external point of a circle two tangents can be
drawn to the circle.
Q
Major sector
O
P
Secan
t
Minor
sector
Q
Radiu
s
Tangen
t
4.
Minor
segment
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6. Secant - A line which intersects the circle at two distinct points is called a Secant.
Properties of Circle
1.
Two circles are congruent if and only it they have equal radii.
2.
Equal chords of a circle subtend equal angles at the center.
(Theorem 10.1)
Conversely if the angles subtended by the chords at the center are equal, then chords
are equal.
(Theorem 10.2)
3.
The perpendicular drawn from the center of a circle to a chord bisects the chord.
(Theorem 10.3)
Conversely the line drawn from the center of a circle to bisect a chord is perpendicular to
the chord.
(Theorem10.4)
4.
There is one and only one circle passing through three given non – collinear points.
(Theorem 10.5)
5.
Equal chords of a circle are equidistant from the center.
(Theorem 10.6)
Conversely Chords equidistant from the center of a circle are equal in length.
(Theorem 10.7)
6.
If two arcs of a circle are congruent, then their corresponding chords are equal.
Conversely if two chords of a circle are equal, then their corresponding arcs are
congruent.
7.
Congruent arcs of a circle subtends equal angle at the center.
8.
C
The angle subtended by an arc at the center is double the angle
subtended by it at any point on the remaining part of the circle.
O
2
B
A
(Theorem 10.8)
9.

D
C
Angles in the same segment of a circle are equal. (Theorem 10.9)
O
B
A
10.
Angle in a semicircle is a right angle.
11.
If a line-segment joining two points subtends equal angles at two other points lying on
the same side of the line containing the line-segment, the four points lie on a circle. i.e.
concyclic.
(Theorem 10.10)
12.
The sum of the either pair of the opposite angles of a cyclic
quadrilateral* is 180°.
A
D
B
i.e.
A + C = 180°
O
(Theorem 10.11)
Conversely if the sum of a pair of opposite angles of a quadrilateral is 180°
(i.e. supplementary), the quadrilateral is cyclic.
(Theorem 10.12)
C
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Address : 61, First Floor, Chanakya Plaza, New CG Road, Chandkheda, Ahmedabad
Contact : 96017 55017 , 90999 66717
www.abhigyanam.com
SOME IMPORTANT QUESTIONS:
1
Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to
each other and are on the same side of its centre as shown in figure 1. If the distance
between AB and CD is 3 cm, find the radius of the circle.
2
In figure 2, AB = CB and O is the center of the circle. Prove that BO bisects ABC.
3
In figure 3, OD is perpendicular to chord AB of a circle whose center is O.
If BC is a diameter, prove that CA = 2OD.
B
A
l
C
B
D
M
O
5
6
7
8
A
D
x
D
E
D
C
Fig -6
11
12
A
B
O
10
Fig -3
Fig -2
PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm. If
PQ = 16 cm and RS = 12 cm, find the distance between PQ and RS, if they lie.
(i)
On the same side of the center O (ii)
On opposite side of the center O.
A line segment AB is of length 5 cm. Draw a circle of radius 4 cm passing through A and
B. Can you draw a circle of radius 2 cm passing through A and B? Give reasons in
support of your answer.
AB and CD are equal chords of a circle whose center is O. When produced, these chords
meet at E as shown in figure 6. Prove that EB = ED.
In figure 7, AB and AC are two equal chords of a circle whose center is O. If OD  AB and
OE  AC, prove that ADE is an isosceles triangle.
BC is a chord of a circle with center O. A is a point on an arc BC as shown in figure 8 (i) &
(ii). Prove that
(i)
BAC + OBC = 90°, if A is the point on the major arc.
(ii)
BAC - OBC = 90°, if A is the point on the minor arc.
A
9
C
C
Fig -1
B
O
O
A
4
D
D
A
E
C
Fig -7
B
O
z
O
B
A
xD
D
B
y
y
z
y
t
O
y
C
C
Fig -8
Two diameters of a circle intersect each other at right angles. Prove that the
quadrilateral formed by joining their end points is a square.
Two circles intersect each other at points A and B. If AP and AW be the respective
diameters of the two circles, prove that PBW is a line.
Two congruent circles intersect each other at points P and Q. A line through P meets the
circles in A and B. Prove that QA = QB.
D and E are, respectively, the points on equal sides AB and AC of an isosceles triangle
ABC such that AD = AE. Prove that points B, C, E and D are con-cyclic.
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Contact : 96017 55017 , 90999 66717
www.abhigyanam.com
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________________________________________________________________________________________
Address : 61, First Floor, Chanakya Plaza, New CG Road, Chandkheda, Ahmedabad
Contact : 96017 55017 , 90999 66717
www.abhigyanam.com