### Congruent circles

```Chapter-12 CIRCLES
MATH CLASS-9
Module Objectives
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Define circle.
Identify and state the property of chord of a circle.
Identify central angle and inscribed angle.
State the theorem on angle property of the circle.
Prove the theorem logically.
Solve problems and riders based on the properties of the circle.
Introduction
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Circles are one of the interesting figures in geometry.
The wheels of the carts, automobiles and trains are circular.
What is Circle?
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A closed curve, every point of which is equidistant from a given fixed point.
This fixed point (O) is called the centre of the circle.
A circle is the locus of a moving point in a plane such that it is at a constant distance from the given fixed
point.
Locus is the path traced by a moving point
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The line segment joining the centre to any point on the circle.
OA and OB are the radii of the given circle.
The radii of a circle are always equal.
i.e: OA == OB
Circumference/Perimeter is the Distance around the circle.
Chord & Diameter
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CHORD-A line segment with its end points lying on the circle.
Here AB,PQ and XY are the chords.
DIAMETER-A line segment passing through the centre of the circle
and has its end points on circle.
Diameter is the longest chord of a circle.(PQ)
Length of a diameter is twice as its radius.
Here PQ=OP+OQ
D=2R or Radius = Diameter/2 i.e: R = D/2
A
B
O
P
X
Y
Q
Concentric and Congruent Circles
Circles C1,C2,C3 have the
same center O and different
and OC(2.5cm).
Circles C1,C2,C3 have
different centres P,Q and
PA=QB=RC=1.5 cm.
Concentric circles are circles
with same centre and different
Congruent circles are
different centres.
ARC OF A CIRCLE
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An arc is a `Part’ of a Circle.
The chord divides the circle into two parts,One smaller than the other.
The smaller part is called the minor arc which is denoted as AXB.
The greater part is called the major arc which is denoted as AYB.
AXB and AYB are conjugate arcs.They lie on either sides of chord AB.
If a chord is a diameter ,it divides the circle into two arcs of same size.Each is a semi circle.
Here CXD and CYD are semicircles.
Segment of a Circle
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The part of the circular region included by an arc and the chord is called a segment.
The region bounded by the chord AB and the major arc APB is called the major segment.
The region bounded by the chord AB and the minor arc AQB is called the minor segment.
The region bounded by the diameter and the arc is called the semi circular region.
Do it Yourself 12.1(a)
1.Observe the figure and represent the following :
(a) OP
(b) SO
(c) SQ
(d) SPQ
(f) Conjugate of SPQ.
2. What is the length of the biggest chord of a circle of radius 4 cm?
3. Draw a circle with diameter 7cm.
4. Draw a circle with centre O.Draw two diameters and label their end points A,B,C and D.
Draw the chords that connect the end points of the diameters.Name the following:
(a) Four pairs of congruent triangles.
(b) a pair of parallel chords.
( c ) rectangle.
(d) semicircle.
5. Draw a circle with centre ‘O’ and radius 2 cm . For this circle, draw a concentric and a congruent circle.
Properties of Chord in a Circle
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In each of the circles given below,PG is the chord and OA is perpendicular to PQ.In each case PA=QA.
Property: In a circle the perpendicular from the centre to the chord , bisects the chord
Properties of chord of a circle
Property: Equal chords of a circle are equidistant from the centre
Here the two chords AB and RS are equidistant from the
centre(OD=OF).Hence AB=RS.
In a circle if BC is the chord and OA is perpendicular to BC,
OB²=OA²+AB²
r²=d²+AB²
r² = d²+l²
d= perpendicular distance of chord from centre
l= length of the chord/2
Do it Yourself 12.1(b)
1. In a circle, with centre O ,the chords and their distances from the centre are given. Arrange them
according to the increasing order of their length.
S.No
Name of Chord
Distance
1
PQ
4.6 cm
2
AB
3.6 cm
3
XY
1 cm
4
CD
2.1 cm
5
MN
0 cm
2. The length of chords AB,PQ,MN,DE and XY are 5.1 cm,2.9 cm,6.3 cm,4.5 cm and 5.4 cm.Arrange
them in decreasing order of their distance from the centre of the circle.
3. Chord PQ = Chord AB. If PQ is at a distance of 3 cm from the centre of the circle at what distance is
chord AB from the centre of the circle.
4 .In the given figure, OA and OB are the radii of the circle ,AB is the chord. OP is perpendicular to AB.
Prove that AP = PB.
ANGLE PROPERTIES OF A CIRCLE
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Angles are of paramount importance in geometry and also in real life.
Surveyors,Engineers,Navigators,Astronomers ,Scientists and many other people use the measurements of
angles.
In Circle 4,the vertex of the angle lies with the centre of the
circle.Such an angle is called central angle.The arms of the
In Circle 1,the vertex of the angle lies on the circle and its
arms intersect the circle at two points.Such an angle is
called inscribed angle.
Think!
How many central angles can
be drawn to intercept the
circleat A and B?
Think!
In circles 2 and 3 can angle
 ACB be called an
inscribed angle?Why?
Think!
How many angles can be
inscribed in a circle by the
same arc?
Relation between Central Angle and Inscribed Angle
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In the circles given below, O is the centre of the circle and the arc AXB subtends angle  AOB at the
centre and  ACB on the remaining part.
Measure angles  AOB and  ACB seperately .It can be found that ,  AOB = 2  ACB.
The angle subtended by an arc at the centre is double the angle subtended by the same arc at any
point on the remaining part of the circle.
Angles in a Segment
Measure each of the angles in the following figures and record them in the table.
From the above diagram we can conclude that:
• Angle in the major segment is an acute angle.
• Angle in the minor segment is an obtuse angle.
• Angle in the semi-circle is a right angle.
• Angles in the same segment are equal.
Angle
Segment
1
Major
2
Minor
3
Semicircle
4
Semicircle
5
Minor
6
Minor
Measure
Angles in a Segment
Solution:
1.
 X = 2 × 30˚
x = 60˚
 y = 30˚
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3.
4.
(angle at the centre is twice the angle at any point on the circle.)
(angles in the same segment are equal)
X = ½
× 105˚
 X = 52.5˚
(angle at any point on the circle is half the angle at centre)
 X=½ ×
 X = 40˚
80˚
(angle at any point on the circle is half the angle at the centre)
Angles in a Segment
Example-2: Find the angles of the triangle ACB.
Solution :
 ABC = 30˚
(given)
 ACB = 90˚ (angle in a semicircle)
Hence  CAB = 180˚ - ( 30˚ + 90˚ )
= 180˚ - 120˚
 CAB = 60˚
Do it yourself 12.1(c)
1.Why angle D is not an inscribed angle?
3. a) Name four central angles with respect to adjoining figure.
b) Name two inscribed angles.
c) Name two angles that subtend BC.
d) What angle subtends CD?
e) What kind of triangle is DOC?
f) Name three chords.
g) Which chord is a diameter?
2.Why angle E is not a central angle?
Do it yourself 12.1(c)
a) Name the central angle subtended by AE.
b) Name the central angle subtended by BC.
c) Name the inscribed angle subtended by BC.
d) Name the central angle subtended by CDE.
e) Name two chords that are not diameters.
f) Name a chord that is a diameter.
g) Name the arc subtended by BOA.
h) Name the arc subtended by DEB.
5. Write the value of x in each of the following cases.
Do it yourself 12.1(c)
6. Draw a rough diagram for each of the following:
a) AB is a chord of a circle , with centre O.If  OAB = 50˚,Find OBA
b) RS is a chord of a circle with centre O.If  ROS = 15˚,Find ORS
7. a) What kind of triangle is AOB ?
b) What can you say about angles  A and
c) What kind of angles are 1 and 2 ?
B ?
Angles in a Segment
Data: `O’ is the centre of the circle.AXB is the arc.AOB is the angle
subtended by the arc AXB at the centre.ACB is the angle subtended by
the arc AXB at a point on the remaining part of the circle.
To Prove: AOB = 2  ACB
Construction : Join CO and produce it to D.
Statement
Reason
1. OA = OC
Radii of the same circle are equal.
2. OCA = OAC
Angles opposite to equal sides are equal.
3.In ∆AOC, AOD =
OCA + OAC
4. AOD =  OCA +
OCA
Substituting
5.  AOD = 2  OCA
6.Similarly in ∆ BOC ,
Exterior angle of a triangle = sum of interior
opposite angles.
 OAC by OCA from statement .
 BOD
= 2  OCB
7. AOD + BOD = 2 OCA + 2 OCB = 2(OCA+OCB)
Taking out 2 as common.
8.  AOB = 2  ACB
Since AOD + BOD = AOB ,OCA +OCB = ACB
Angles in a Segment
Example-1 : Prove that angle in a semicircle is a right angle.
Data: AOB is the diameter.
 ACB is angle in the semicircle.
To prove: ACB = 90°
Proof: 1)  AOB = 180°
( AOB is a straight line )
2)  ACB = ½ AOB (angle at any point on the circle is half the angle at centre.)
3)  ACB = ½ × 180° (from 1)
4)  ACB = 90°
Example-2 : From the adjoining diagram , prove that ∆APC and ∆DPB are equiangular.
To prove : ∆ APC and ∆DPB are equiangular.
Proof : In ∆ APC and ∆DPB
 APC =  BPD (Vertically opposite angles)
 ACP =  ABD (Angles in the same segment are equal)
 PAC =  PDB (Angles in the same segment are equal)
Hence ∆ APC and ∆DPB are equiangular.
C
A
B
Do it yourself 12.1(d)
1. In the given figure , prove that  PRQ = PSQ
2. In the figure given below AC and BC are diameters of two circles intersecting at C and D. Show that A,D,B are
collinear.
3.Two chords AB and CD of a circle intersect at P.If BP =PD, show that AC ll BD.
4.In the adjoining figure D is a point outside the circle and  ACB = 40°.Show that
Do it yourself 12.1(d)
5. In the given figure if  ASC = 160° and  ABC = 80°.Prove that ‘S’ is the circumcentre of the ∆ABC.
6. PQ is a diameter of a circle with the centre O, and R is any other point on the circle and  RPO =
25°.Calculate  OQR.
7. O is the circumcentre of ∆ABC.If  ABC = 32°.Calculate AOC.
8. AC and BD are chords of a circle which intersect at X.If  ACD = 35° and  BCA = 20 °.Calculate (i)  ABD and
(i)  BDA.
9.’O’ is the circumcentre of ∆ABC.If AB = BC and  BAC = 50°.Calculate  ABC,  AOC and OAC.
10.AOB is a diameter of a circle centre O.If,C is a point on the circle and  BCO = 60°.Calculate  OCA ,  OAC
and  AOC.
11.’O’ is the circumcentre of ∆PQR.If PQR = 40° and  RPQ = 50 °.Calculate POQ.
END OF CHAPTER
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