1701832667764.QP PS CB IX Math Circles

PODAR INTERNATIONAL SCHOOL (CBSE)
Practice sheet
Std: IX Chapter: 10 Circles
Subject: Mathematics
Q.1 AC and BD are chords of a circle which bisect each other. Prove that:
(i) AC and BD are diameters
(ii) ABCD is a rectangle
Q.2 OD is perpendicular to chord AB of a circle whose centre is O. If BC is A D B

a diameter, prove that CA = 2OD.
O
C
Q.3 In the figure, diameter AB and a chord AC have a common end point
A D C
A. If the length of AB is 20 cm and of AC is 12 cm, how far is AC
from the centre of the circle?
O
B
Q.4 AB and AC are two equal chords of a circle. Prove that the bisector of BAC passes
through the center of the circle.
Q.5 AB and AC are two chords of a circle of radius r such that AB = 2AC. If and are the
distances of AB and AC from the centre, prove that
Q.6 If two circles intersect at two points, prove that their centres lie on the perpendicular bisector
of the common chord.
Q.7 If a line segment joining mid-points of two chords of a circle passes through the centre,
prove that the two chords are parallel.
CB/IX/2122 Circles page 1 of 7

Q.8 In the adjoining figure, AOC is a diameter of the circle and
1
Arc AXB  arc BYC Find ∠BOC.
2
Q.9 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
Q.10 A, B and C are three points on a circle. Prove that the perpendicular bisectors of AB, BC
and CA are concurrent.
Q.11 Two circles with centers O and intersect at two points A and B. A line PQ is drawn
parallel to through A (or B) intersecting the circles at P and Q. Prove that PQ =
Q.12 Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius
5 m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma.
If the distance between Reshma and Salma, and Salma and Mandip is 6 m each, what is the
distance between Reshma and Mandip?
Q.13 Two equal chords AB and CD of a circle whose centre is O when produced meet at a point
P. Prove that PB = PD and AP = CP
(Hint: Draw perpendiculars from O to AB and CD)
Q.14 In the adjoining figure, O is the centre of the circle, BD = OD and

CD AB. Find CAB.

Q.15 Two chords AB and AC of a circle subtend angles equal to 150o and 90o respectively at the
centre. Find BAC, if AB and AC lie on the opposite sides of the centre.
Q.16 The circumcentre of a triangle ABC is O. Prove that OBC+ BAC = 90o.
Q.17 ABCD is such a quadrilateral that A is the center of the circle passing through B, C and D.
Prove that ∠CBD + ∠CDB = 1 ∠BAD.

2
Q.18 In the given figure, AB and CD are two chords of a circle with centre O intersecting each
other at E.
Prove that AEC = 1 (angle subtended by arc CXA at centre + angle subtended by arc
2
DYB at centre).
Q.19
A circle has radius cm. It is divided into two segments by a chord or length 2 cm.
Prove that the angle subtended by the chord at a point in major segment is 45o.
Q.20 In the given figure, BAD=65o, ABD=70o and BDC=45o.

(i) Prove that AC is a diameter of the circle.
(ii) Find ACB.

Q.21 In the adjoining figure, ED is a chord parallel to the
diameter AC of the circle ABCDE. If ∠ CBE = 63o,
calculate ∠DEC.
Q.22 In the adjoining figure, O is the center of the circle.

If ∠BCO = 30o, find x and y.
Q.23 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 concyclic.
Q.24 A quadrilateral ABCD is inscribed in a circle such that AB is diameter and ADC=130o.
Find ∠BAC.
Q.25 In the adjoining figure, AB is a diameter of the circle and C,

D, E are any three points on the semicircle. Find the value of
∠ACD +∠BED.
Q.26 In the adjoining figure, AB is a diameter of the circle with

centre O. If ∠ADC =130o and chord BC = chord BE,
find ∠CBE.

Q.27 If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides, prove
that quadrilateral so formed is cyclic.
Q.28 If a pair of opposite sides of a cyclic quadrilateral is equal. Prove that its diagonals are also
equal.
Q.29 ABCD is a parallelogram, A circle through A and B is so drawn that it intersects AD at P

and BC at Q. Prove that points P, Q, C and D are concyclic.
Q.30 If the bisectors of opposite angles of a cyclic quadrilateral intersect the circle circumscribing
it at the points P and Q, Prove that PQ is a diameter of the circle.
Q.31 ABCD is a cyclic trapezium with AD || BC. If B = 70, determine other three angles of the
trapezium.
Q.32 ABC is an isosceles triangle with AB = AC. A circle through B and C intersects AB and AC
at D and E respectively. Prove that BC || DE.
Case Study Questions

A farmer has a circular garden as shown in the picture. He has different type of trees and
flower plants in his garden. In the garden, there are two mango trees A and B at a distance of
10 m(AB=10 m).Similarly he has two Ashok trees at the same distance of 10 m as shown at
C and D. AB subtends ∠AOB=120° at the centre O, The perpendicular distance of AC from
centre is 5 m the radius of the circle is 13 m.

(a) What is the measure of ∠COD?
i. 60°
ii. 120°
iii. 80°
iv. 100°
(b) What is the distance between Mango Tree A and Ashok tree C?
i. 12 m
ii. 24 m
iii. 13 m
iv. 14 m
(c) What is the value of angle ∠OAB?
i. 60°
ii. 120°
iii. 30°
iv. 90°
(d) What is the value of ∠ODC?
i. 60°
ii. 120°
iii. 30°
iv. 90°
Case Study 2
Government of India is working regularly for the growth of handicapped persons. For this,
STD booth situated at A, B and C as shown in the figure, are opened which are operated by
handicapped person. These three booths are equidistant from each other as shown in the
figure.

(a) Which type of triangle is ABC?
i. equilateral triangle
ii. isosceles triangle
iii. right angled triangle
iv. scalene triangle
(b) The measure of ∠ABC is
i. 45°
ii. 60°
iii. 30°
iv. 90°
(c) If AB = 6cm then the value of BC + CA is
i. 10
ii. 12
iii. 14
iv. 16
(d) The measure of angle ∠BOC =
i. 90°
ii. 100°
iii. 120°
iv. 50°
(e) The value of ∠OBC + ∠OCB
i. 60°
ii. 30°
iii. 45°
iv. 90°
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PODAR INTERNATIONAL SCHOOL (CBSE)

Q.2 OD is perpendicular to chord AB of a circle whose centre is O. If BC is A D B

distances of AB and AC from the centre, prove that

CB/IX/2122 Circles page 1 of 7

Q.14 In the adjoining figure, O is the centre of the circle, BD = OD and

CB/IX/2122 Circles page 2 of 7

Prove that ∠CBD + ∠CDB = 1 ∠BAD.

Q.20 In the given figure, BAD=65o, ABD=70o and BDC=45o.

CB/IX/2122 Circles page 3 of 7

Q.22 In the adjoining figure, O is the center of the circle.

Q.25 In the adjoining figure, AB is a diameter of the circle and C,

Q.26 In the adjoining figure, AB is a diameter of the circle with

CB/IX/2122 Circles page 4 of 7

Q.29 ABCD is a parallelogram, A circle through A and B is so drawn that it intersects AD at P

Case Study Questions

CB/IX/2122 Circles page 5 of 7

CB/IX/2122 Circles page 6 of 7

CB/IX/2122 Circles page 7 of 7

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