RD SHARMA SOLUTION CHAPTER -15 Circles| CLASS 9TH MATHEMATICS-EDUGROWN

Chapter 15 – Circles Exercise 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) threeQuestion 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 180^o.Solution 2

(i) T

(ii) T

(iii) T

(iv) F

(v) T

(vi) T

(vii) F

(viii) T

Chapter 15 – Circles Exercise 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 ONDOD=OB=5cm             (radii of same circle)

 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 15Two 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 15Draw 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 Exercise 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 Exercise 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(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

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

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

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

Solution 3(ix)

Question 3(x)

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

Solution 3(x)

Question 3(xi)

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

Solution 3(xi)

Question 3(xii)

Solution 3(xii)

Question 4

Solution 4

Question 5

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

Solution 5

Question 6

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

Solution 6

Question 7

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

Solution 7

Question 8

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 8

Question 9

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

Solution 9

Question 10

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

Solution 10

Question 11

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

Solution 11

Chapter 15 – Circles Exercise Ex. 15.5

Question 1

In fig., ΔABC is an equilateral triangle. Find m∠BEC.

Solution 1

Question 2

In fig., ΔPQR is an isosceles triangle with PQ = PR and m∠PQR = 35°. find m∠QSR and m∠QTR.

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(i)

Solution 18(i)

Question 18(ii)

Solution 18(ii)

Question 18(iii)

Solution 18(iii)

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

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

Solution 23

Question 24

Solution 24

Question 25

Prove that the centre of the circle circumscribing the cyclic rectangle ABCD is the point of intersection of its diagonals.Solution 25

Let O be the centre of the circle circumscribing the cyclic rectangle ABCD. Since ABC = 90^o 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 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

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 29