# Chapter 15 : Circles - Rd Sharma Solutions for Class 9 Maths CBSE

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## Chapter 15 - Circles Excercise Ex. 15.1

Question 1

Fill in the blanks:

(i) All points lying inside/outside a circle are called ...... points/ ... points.

(ii) Circles having the same centre and different radii are called ... circles.

(iii) A point whose distance from the centre of a circle is greater than its radius lies in ... of the circle.

(iv) A continuous piece of a circle is ... of the circle.

(v) The longest chord of a circle is a ... of the circle.

(vi) An arc is a ... when its ends are the ends of a diameter.

(vii) Segment of a circle is the region between an arc and ... of the circle.

(viii) A circle divides the plane, on which it lies, in .... parts.

Solution 1

(i) interior/exterior

(ii) concentric

(iii) the exterior

(iv) arc

(v) diameter

(vi) semi-circle

(vii) centre

(viii) three

Question 2

Write the truth value (T/F) of the following with suitable reasons:

(i) A circle is a plane figure.

(ii) Line segment joining the centre to any point on the circle is a radius of the circle.

(iii) If a circle is divided into three equal arcs each is a major arc.

(iv) A circle has only finite number of equal chords.

(v) A chord of a circle, which is twice as long is its radius is a diameter of the circle.

(vi) Sector is the region between the chord and its corresponding arc.

(vii) The degree measure of an arc is the complement of the central angle containing the arc.

(viii) The degree measure of a semi-circle is 180o.

Solution 2

(i) T

(ii) T

(iii) T

(iv) F

(v) T

(vi) T

(vii) F

(viii) T

## Chapter 15 - Circles Excercise Ex. 15.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Give a method to find the centre of a given circle.

Solution 4

Steps of construction:

(1) Take three point A, B and C on the given circle.

(2) Join AB and BC.

(3) Draw the perpendicular bisectors of chord AB and BC which interesect each other at O.

(4) Point O will be the required circle because we know that the perpendicular bisector of a chord always passes through the centre.

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord form the centre?

Solution 11

Distance of smaller chord AB from centre of circle = 4 cm.
OM = 4 cm

In OMB
In OND

So, distance of bigger chord from centre is 3 cm.
Question 12

Solution 12

Question 13

Solution 13

Question 14

Prove that two different circles cannot intersect each other at more than two points.

Solution 14

Suppose two different circles can intersect each other at three points then they will pass through the three common points but we know that there is one and only one circle with passes through three non-collinear points, which contradicts our supposition.

Hence, two different circles cannot intersect each other at more than two points.

Question 15
Two chords AB and CD of lengths 5 cm and 11cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.
Solution 15
Draw OM  AB and ON  CD. Join OB and OD

(Perpendicular from centre bisects the chord)
Let ON be x, so OM will be 6 - x
In MOB
In NOD

We have OB = OD             (radii of same circle)
So, from equation (1) and (2)

From equation (2)

So, radius of circle is found to be  cm.

## Chapter 15 - Circles Excercise Ex. 15.3

Question 1

Three girls Ishita, Isha and Nisha are playing a game by standing on a circle of radius 20 m drawn in a park. Ishita throws a ball to Isha, Isha to Nisha, Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is 24 m each, what is the distance between Ishita and Nisha?

Solution 1

Question 2

A circular park of radius 40 m is situated in a colony. Three boys Ankur, Amit and Anand are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

Solution 2

## Chapter 15 - Circles Excercise Ex. 15.4

Question 1

In fig., O is the centre of the circle. If APB = 50°, find AOB and OAB.

Solution 1

Question 2

In fig., O is the centre of the circle. Find BAC.

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

If O is the centre of the circle. Find the value of x in the following figure:

Solution 9

Question 10

Solution 10

Question 11

If O is the centre of the circle. Find the value of x in the following figure:

Solution 11

Question 12

If O is the centre of the circle. Find the value of x in the following figure:

Solution 12

Question 13

If O is the centre of the circle. Find the value of x in the following figure:

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

In fig., O is the centre of the circle, BO is  the bisector of ABC. Show that AB = AC.

Solution 16

Question 17

In fig., O and O' are centres of two circles intersecting at B and C. ABD is straight line, find x.

Solution 17

Question 18

In fig., if ACB = 40°, DPB = 120°, find CBD.

Solution 18

Question 19

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Solution 19

Question 20

In fig., it is given given that O is the centre of the circle and AOC = 150°. Find ABC.

Solution 20

Question 21

In fig., O is the centre of the circle, prove that x = y + z.

Solution 21

Question 22

in fig., O is the centre of a circle and PQ is a diameter. If ROS = 40°, find. RTS.

Solution 22

## Chapter 15 - Circles Excercise Ex. 15.5

Question 1

In fig., ΔABC is an equilateral triangle. Find mBEC.

Solution 1

Question 2

In fig., ΔPQR is an isosceles triangle with PQ = PR and mPQR = 35°. find mQSR and mQTR.

Solution 2

Question 3

In fig., O is the centre of the circle. If BOD = 160°, find the values of x and y.

Solution 3

Question 4

In fig., ABCD is a cyclic qudrilateral. If BCD = 100° and ABD = 70°, find ADB.

Solution 4

Question 5

If ABCD is a cyclic quadrilateral in which AD  BC. Prove that B = C.

Solution 5

Question 6

In fig., O is the centre of the circle. find CBD.

Solution 6

Question 7

In fig., AB and CD are diameters of a circle with centre O. If OBD = 50°, find AOC.

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

In fig., O is the centre of the circle and DAB = 50. calculate the values of x and y.

Solution 11

Question 12

In fig., if BAC = 60°, and BCA = 20°, find ADC.

Solution 12

Question 13

In fig., if ABC is an equilateral triangle. Find BDC and BEC.

Solution 13

Question 14

In fig., O is the centre of the circle. If CEA = 30°, find the values of x, y and z.

Solution 14

Question 15

In fig., BAD = 78°, DCF = x° and DEF = y° find the values of x and y.

Solution 15

Question 16

Solution 16

Question 17

In fig., ABCD is cyclic qudrilateral. Find the value of x.

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

In fig., ABCD is cyclic quadrilaterial in which AC an BD are its diagonals. If DBC = 55° and BAC = 45°, find BCD.

Solution 25

Question 26

Solution 26

Question 27

Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of intersection of its diagonals.

Solution 27

Let O be the centre of the circle circumscribing the cyclic rectangle ABCD. Since ABC = 90o and AC is a chord of the circle, so, AC is a diameter of the circle. Similarly, BD is a diameter.

Hence, point of intersection of AC and BD is the centre of the circle.

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to its opposite vertex is half of the hypotenuse.

Solution 31

## Chapter 15 - Circles Excercise 15.109

Question 1

If the length of a chord of a circle is 16 cm and is at a distance of 15 cm from the centre of the circle, then the radius of the circle is

(a) 15 cm

(b) 16 cm

(c) 17 an

(d) 34 cm

Solution 1

Question 2

Solution 2

## Chapter 15 - Circles Excercise 15.110

Question 1

(a) 60°

(b) 45°

(c) 30°

(d) 15°

Solution 1

Question 2

Solution 2

Question 3

A chord of length 14 cm is at a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is

(a) 12 cm

(b)) 14 cm

(c) 16 cm

(d) 18 cm

Solution 3

Question 4

One chord of a circle is known to be 10 cm. The radius of this circle must be

(a) 5 cm

(b) greater than 5 cm

(c) greater than or equal to 5 cm

(d) less than 5 cm

Solution 4

Question 5

ABC is a triangle with B as right angle, AC = 5 cm and AB = 4 cm. A circle is drawn with A as centre and AC as radius. The length of the chord of this circle passing through C and B is

(a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

In a circle, the major arc is 3 times the minor arc. The corresponding central angles and the degree measures of two arcs are

(a) 90° and 270°

(b) 90° and 90°

(c) 270° and 90°

(d) 60° and 210°

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

If two diameters of a circle intersect each other at right angles, then quadrilateral formed joining their end points is a

(a) rhombus

(b) rectangle

(c) parallelogram

(d) square

Solution 11

Question 12

Solution 12

Question 13

The chord of a circle is equal to its radius. The angle subtended by this chord at the minor of the circle is

(a) 60

(b) 750

(c) 1200

(d) 1500

Solution 13

## Chapter 15 - Circles Excercise 15.111

Question 1

Solution 1

Question 2

Solution 2

Question 3

The greatest chord of a circle is called its

(b) secant

(c) diameter

(d) none of these

Solution 3

The greatest chord of the circle is diameter of the circle.

Hence, correct option is (c).

Question 4

Angle formed in minor segment of a circle is

(a) acute

(b) obtuse

(c) right angle

(d) none of these

Solution 4

Angle formed in a minor segment is always a obtuse angle.

Hence, correct option is (b).

Question 5

Number of circles that can be drawn through three non-collinear points is

(a) 1

(b) 0

(c) 2

(d) 3

Solution 5

Three non-collinear points make a triangle and there is only one circle that can pass through all three points,

i.e. circumcircle of that triangle.

Hence, correct option is (a).

Question 6

In figure, if chords AB and CD of the circle intersect each other at right angles, then x + y =

(a) 450

(b) 600

(c) 750

(d) 900

Solution 6

Question 7

Solution 7

Question 8

In figure, chords AD and BC intersect each other at right angles at a point P. If

(a) 350

(b) 450

(c) 550

(d) 650

Solution 8

Question 9

Solution 9

## Chapter 15 - Circles Excercise 15.112

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

In a circle of radius 17 cm, two parallel chords are drawn on opposite side of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is

(a) 34 cm

(b) 15 cm

(c) 23 cm

(d) 30 cm

Solution 5

Question 6

Solution 6