Chapter-12 CIRCLES. MATH CLASS-9. Module Objectives. Define circle . Define radius,circumference,arc,line segment,chord . 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 .
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i.e: OA == OB
and has its end points on circle.
D=2R or Radius = Diameter/2 i.e: R = D/2
Circles C1,C2,C3 have the same center O and different radiiOA(1.2cm),OB(1.8cm)
Concentric circles are circles with same centre and different radii.
Circles C1,C2,C3 have different centres P,Q and R.But they have same radii,
Congruent circles are circles with same radii but different centres.
1.Observe the figure and represent the following :
(e ) Shaded portion
(f) Conjugate of SPQ.
2. Whatis the length of the biggestchord of a circle of radius 4 cm?
3. Draw a circlewithdiameter7cm.
4. Draw a circlewith centre O.Drawtwodiameters and label their end points A,B,C and D.
Draw the chordsthatconnect the end points of the diameters.Name the following:
(a) Four pairs of congruent triangles.
(b) a pair of parallelchords.
( c ) rectangle.
5. Draw a circlewith centre ‘O’ and radius 2 cm . For thiscircle, draw a concentric and a congruent circle.
Property: In a circle the perpendicular from the centre to the chord , bisects the chord
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,
r² = d²+l²
d= perpendicular distance of chord from centre
l= length of the chord/2
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.
2. The length of chords AB,PQ,MN,DE and XY are 5.1 cm,2.9cm,6.3cm,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.
circle.Such an angle is called central angle.The arms of the
central angle are its radii.
arms intersect the circle at two points.Such an angle is
called inscribed angle.
How many central angles can
be drawn to intercept the circleat A and B?
In circles 2 and 3 can angle ACB be called an inscribed angle?Why?
How many angles can be inscribed in a circle by the same arc?
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.
Measure each of the angles in the following figures and record them in the table.
X = 2 × 30˚ (angle at the centre is twice the angle at any point on the circle.)
x = 60˚
y = 30˚ (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 × 80˚
X = 40˚ (angle at any point on the circle is half the angle at the centre)
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˚
1.Why angle D is not an inscribed angle? 2.Why angle E is not a central 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?
4. In the adjoining figure,
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.
6. Draw a rough diagram for each of the following:
a) AB is a chord of a circle , with centre O.IfOAB = 50˚,Find OBA
b) RS is a chord of a circle with centre O.IfROS = 15˚,Find ORS
7. a) What kind of triangle is AOB ?
b) What can you say about angles A and B ?
c) What kind of angles are 1 and 2 ?
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.
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.
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 ADB < 40°.
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°.CalculateOQR.
7. O is the circumcentre of ∆ABC.If ABC = 32°.CalculateAOC.
8. AC and BD are chords of a circle which intersect at X.IfACD = 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°.CalculateOCA , OAC and AOC.
11.’O’ is the circumcentre of ∆PQR.IfPQR = 40° and RPQ = 50 °.Calculate POQ.