NCERT Exemplar Class 9 Maths

CHAPTER- 10

CIRCLE

(A) Main Concepts and Results

Circle, radius, diameter, chord, segment, cyclic

quadrilateral.

• Equal chords of a circle (or of congruent circles)

subtend equal angles at the centre,

• If the angles subtended by the chords of a circle (or

of congruent circles) at the centre (or centres) are
equal, then the chords are equal,

• The perpendicular drawn from the centre of the circle

to a chord bisects the chord,

• The line drawn through the centre of a circle bisecting

a chord is perpendicular to the chord,

• There is one and only one circle passing through

three given non-collinear points,

• Equal chords of a circle (or of congruent circles) are

equidistant from the centre (or centres),

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• Chords equidistant from the centre of a circle are

equal in length,

• If two chords of a circle are equal, then their

corresponding arcs are congruent

and conversely, if two arcs are congruent, then their

corresponding chords are equal,

• Congruent arcs of a circle subtend equal angles at the

centre,

• The angle subtended by an arc at the centre is double

the angle subtended by it at any point on the
remaining part of the circle,

• Angles in the same segment of a circle are equal,

• 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, then the four points
are con cyclic,

• The sum of either pair of opposite angles of a cyclic

quadrilateral is 180º,

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• If the sum of a pair of opposite angles of a

quadrilateral is 180º, the quadrilateral is cyclic.

EXERCISE 10.1

1. AD is a diameter of a circle and AB is a chord. If AD

= 34 cm, AB = 30 cm, the distance of AB from the
centre of the circle is :

(A) 17 cm (B) 15 cm (C) 4 cm (D) 8 cm

Solution-

diameter
r = OA = = 17cm
2

1
AM = AB[ Per. from the center of circle to the chord bisect it]
2
1
= ´ 30 = 15cm
2

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in rt DAMO
OM 2 = OA2 - AM 2 (pythagoras theoram)
=17 2 - 152
= (17 - 15)(17 + 15)
= 2 ´ 32 = 64
OM = 64 = 8cm (D) ans.

2. In Fig. 10.3, if OA = 5 cm, AB = 8 cm and OD is

perpendicular to AB, then CD is equal to:

(A) 2 cm (B) 3 cm

(C) 4 cm (D) 5 cm

Solution-

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OC ^ AB
1
Þ AC= AB
2
1
= ´ 8 = 4cm
2

3. If AB = 12 cm, BC = 16 cm and AB is perpendicular

to BC, then the radius of the circle passing through the
points A, B and C is :

(A) 6 cm (B) 8 cm

(C) 10 cm (D) 12 cm

4. In Fig.10.4, if Ð ABC = 20º, then Ð AOC is equal to:

(A) 20º (B) 40º (C) 60º (D) 10º

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5. In Fig.10.5, if AOB is a diameter of the circle and AC

= BC, then Ð CAB is equal to:

(A) 30º (B) 60º

(C) 90º (D) 45º

6. In Fig. 10.6, if Ð OAB = 40º, then Ð ACB is equal to :

(A) 50º (B) 40º (C) 60º (D) 70°

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7. In Fig. 10.7, if Ð DAB = 60º, Ð ABD = 50º, then Ð

ACB is equal to:

(A) 60º (B) 50º (C) 70º (D) 80º

8. ABCD is a cyclic quadrilateral such that AB is a

diameter of the circle circumscribing it and Ð ADC =
140º, then Ð BAC is equal to:

(A) 80º (B) 50º

(C) 40º (D) 30º

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9. In Fig. 10.8, BC is a diameter of the circle and Ð BAO

= 60º. Then Ð ADC is equal to :

(A) 30º (B) 45º

(C) 60º (D) 120º

10. In Fig. 10.9, Ð AOB = 90º and Ð ABC = 30º, then Ð

CAO is equal to:

(A) 30º (B) 45º (C) 90º (D) 60º

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EXERCISE 10.2

Write True or False and justify your answer in each of

the following:

1. Two chords AB and CD of a circle are each at

distances 4 cm from the centre. Then AB = CD.

Solution-

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in DO PC and DOQA
Ð1=Ð2=90
OC= OA=radii
OP = OQ = 4cm
\DOPC @ DOQA by RHS prop.
Þ CP=AQ (Cpct)( ´ 2)
2CP=2AQ
CD=AB[ per. from the center of the circle to the chord bisects it.]
\ True

2. Two chords AB and AC of a circle with centre O are

on the opposite sides of OA. Then Ð OAB = Ð OAC .

Solution-

false.
ÐOAB = ÐOAC if DAOB @ DAOC
or if AB= AC

3. Two congruent circles with centres O and O’

intersect at two points A and B.

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Then Ð AOB = Ð AO’B.

Solution-

4. Through three collinear points a circle can be drawn.

5. A circle of radius 3 cm can be drawn through two

points A, B such that AB = 6 cm.

6. If AOB is a diameter of a circle and C is a point on

the circle, then AC2 + BC2 = AB2

7. ABCD is a cyclic quadrilateral such that ÐA = 90°, Ð

B = 70°, ÐC = 95° and ÐD = 105°.

8. If A, B, C, D are four points such that Ð BAC = 30°

and Ð BDC = 60°, then D is the centre of the circle
through A, B and C.

9. If A, B, C and D are four points such that Ð BAC =

45° and Ð BDC = 45°, then A, B, C, D are con cyclic.

10. In Fig. 10.10, if AOB is a diameter and Ð ADC =

120°, then Ð CAB = 30°.

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EXERCISE 10.3

1. If arcs AXB and CYD of a circle are congruent, find

the ratio of AB and CD.

Solution-

Þ AB= CD [Equal arcs have equal coresponding chords]

AB 1
Þ =
CD 1
Þ AB : CD = 1:1

2. If the perpendicular bisector of a chord AB of a circle

PXAQBY intersects the circle at P and Q, prove that arc
PXA @ Arc PYB.

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Solution-

const. Join PA, PB

to prove

Proof P lies on perpendicular bisector of AB

\ PA=PB

3. A, B and C are three points on a circle. Prove that

the perpendicular bisectors of AB, BC and CA are
concurrent.

Solution-

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Proof- perpendicular bisector of chord of a circle passes

through the center of circle.

\per bisectors of chords AB, BC, and CA passes

through O the center of circle.

Þ perpendicular bisectors arc concurrent at O. [ three

or more lines passing through same point are called
concurrent lines.]

4. AB and AC are two equal chords of a circle. Prove

that the bisector of the angle BAC passes through the
centre of the circle.

5. If a line segment joining mid-points of two chords of

a circle passes through the centre of the circle, prove
that the two chords are parallel.

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6. ABCD is such a quadrilateral that A is the centre of

the circle passing through B, C and D. Prove that Ð CBD

1
+ Ð CDB = Ð BAD
2

7. O is the circumcentre of the triangle ABC and D is

the mid-point of the base BC. Prove that Ð BOD = Ð A.

8. On a common hypotenuse AB, two right triangles

ACB and ADB are situated on opposite sides. Prove that
Ð BAC = Ð BDC.

9. Two chords AB and AC of a circle subtends angles

equal to 90º and 150º, respectively at the centre. Find
Ð BAC, if AB and AC lie on the opposite sides of the
centre.

10. If BM and CN are the perpendiculars drawn on the

sides AC and AB of the triangle ABC, prove that the
points B, C, M and N are con cyclic.

11. If a line is drawn parallel to the base of an isosceles

triangle to intersect its equal sides, prove that the
quadrilateral so formed is cyclic.

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12. If a pair of opposite sides of a cyclic quadrilateral

are equal, prove that its diagonals are also equal.

13. The circumcentre of the triangle ABC is O. Prove

that Ð OBC + Ð BAC = 90º.

14. A chord of a circle is equal to its radius. Find the

angle subtended by this chord at a point in major
segment.

15. In Fig.10.13, Ð ADC = 130° and chord BC = chord

BE. Find Ð CBE.

16. In Fig.10.14, Ð ACB = 40º. Find Ð OAB.

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17. A quadrilateral ABCD is inscribed in a circle such

that AB is a diameter and Ð ADC = 130º. Find Ð BAC.

18. Two circles with centres O and O’ intersect at two

points A and B. A line PQ is drawn parallel to OO’
through A(or B) intersecting the circles at P and Q.
Prove that PQ = 2 OO’.

19. In Fig.10.15, AOB is a diameter of the circle and C,

D, E are any three points on the semi-circle. Find the
value of Ð ACD + Ð BED.

20. In Fig. 10.16, Ð OAB = 30º and Ð OCB = 57º. Find

Ð BOC and Ð AOC.

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EXERCISE 10.4

1. If two equal chords of a circle intersect, prove that

the parts of one chord are separately equal to the parts
of the other chord.

2. If non-parallel sides of a trapezium are equal, prove

that it is cyclic.

3. If P, Q and R are the mid-points of the sides BC, CA

and AB of a triangle and AD is the perpendicular from A
on BC, prove that P, Q, R and D are con cyclic.

4. ABCD is a parallelogram. A circle through A, B is so

drawn that it intersects AD at P and BC at Q. Prove that
P, Q, C and D are concyclic.

5. Prove that angle bisector of any angle of a triangle

and perpendicular bisector of the opposite side if
intersect, they will intersect on the circumcircle of the
triangle.

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6. If two chords AB and CD of a circle AYDZBWCX

intersect at right angles (see Fig.10.18), prove that arc
CXA + arc DZB = arc AYD + arc BWC = semicircle.

7. If ABC is an equilateral triangle inscribed in a circle

and P be any point on the minor arc BC which does not
coincide with B or C, prove that PA is angle bisector of
Ð BPC.

8. In Fig. 10.19, AB and CD are two chords of a circle

1
intersecting each other at point E. Prove that Ð AEC =
2

(Angle subtended by arc CXA at centre + angle

subtended by arc DYB at the centre).

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9. If bisectors of opposite angles of a cyclic

quadrilateral ABCD intersect the circle, circumscribing it
at the points P and Q, prove that PQ is a diameter of
the circle.

10. A circle has radius 2 cm. It is divided into two

segments by a chord of length 2 cm. Prove that the
angle subtended by the chord at a point in major
segment is 45º.

11. Two equal chords AB and CD of a circle when

produced intersect at a point P. Prove that PB = PD.

12. AB and AC are two chords of a circle of radius r

such that AB = 2AC. If p and q are the distances of AB
and AC from the centre, prove that 4q2 = p2 + 3r2.

13. In Fig. 10.20,O is the centre of the circle, Ð BCO =

30°. Find x And Y.

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14. In Fig. 10.21, O is the centre of the circle, BD = OD

and CD ^ AB. Find Ð CAB.

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