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RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 9 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 15 - Circles.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 9 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 9 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.1

Solution 1

(i) interior/exterior

(ii) concentric

(iii) the exterior

(iv) arc

(v) diameter

(vi) semi-circle

(vii) centre

(viii) three

Solution 2

(i) T

(ii) T

(iii) T

(iv) F

(v) T

(vi) T

(vii) F

(viii) T

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.2

Solution 1



Solution 2



Solution 3



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.

Solution 5



Solution 6



Solution 7



Solution 8



Solution 9

Solution 10

Solution 11

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

In OMB In OND OD=OB=5cm             (radii of same circle)   So, distance of bigger chord from centre is 3 cm.

Solution 12



Solution 13



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.

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.

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.3

Solution 1

Solution 2

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.4

Solution 1

Solution 2

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 3(v)

Solution 3(vi)

Solution 3(vii)

Solution 3(viii)

Solution 3(ix)

Solution 3(x)

Solution 3(xi)

Solution 3(xii)

Solution 4



Solution 5

Solution 6

Solution 7

Solution 8



Solution 9

Solution 10

Solution 11

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8



Solution 9



Solution 10



Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18(i)



Solution 18(ii)



Solution 18(iii)



Solution 19



Solution 20



Solution 21



Solution 22



Solution 23

Solution 24

Solution 25



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.

Solution 26



Solution 27



Solution 28



Solution 29



RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.109

Solution 1

begin mathsize 12px style AB space equals space 16 space cm
OC space equals 15 space cm
straight C space is space the space mid minus point space of space AB.
AC space equals space BC space equals 16 over 2 equals 8 space cm
Consider space triangle OCA comma space space space space
OC equals space 15 space cm comma space AC equals 8 space cm
rightwards double arrow OA space equals square root of open parentheses 15 close parentheses squared plus open parentheses 8 close parentheses squared end root space equals space square root of 225 space plus 64 end root equals square root of 289
rightwards double arrow OA space equals space 17 space cm
Hence comma space correct space option space is space left parenthesis straight c right parenthesis.
end style

Solution 2

begin mathsize 12px style AB equals 8 space cm
rightwards double arrow AC equals BC equals 4 space cm
Consider space triangle OCB comma space where space BC space equals space 8 space cm comma space OB space equals space 6 space cm
Now comma space open parentheses OC close parentheses squared space plus space open parentheses BC close parentheses squared space equals space open parentheses OB close parentheses squared
rightwards double arrow left parenthesis OC right parenthesis squared space plus space 4 squared equals 6 squared
rightwards double arrow left parenthesis OC right parenthesis squared space plus space 16 space equals space 36
rightwards double arrow left parenthesis OC right parenthesis squared equals 20
rightwards double arrow OC equals square root of 20 space equals space 2 square root of 5
Hence comma space correct space option space is space open parentheses straight b close parentheses. end style

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.110

Solution 3

begin mathsize 12px style Let space angle BAO space equals straight theta
Consider space triangle OAC comma
sin space straight theta space equals space OC over OA equals fraction numerator begin display style bevelled straight r over 2 end style over denominator straight r end fraction equals 1 half equals sin space 30 degree
rightwards double arrow straight theta space equals space 30 degree
Hence comma space correct space option space is space open parentheses straight c close parentheses. end style

Solution 4

begin mathsize 12px style ABCD space is space straight a space cyclic space Quadrilateral.
Consider space triangle ABD space and space triangle ABC.
Both space are space on space the space same space base space AB space and space angle ADB space and space angle ACB space are space the space angles
in space the space same space segment space AB.
rightwards double arrow angle ADB equals angle ACB equals 30 degree
rightwards double arrow angle BCD space equals space 80 degree plus 30 degree equals 110 degree
In space straight a space cycle space Quadrilateral comma space sum space of space opposite space angles space is space 180 degree.
rightwards double arrow angle straight A plus angle straight C equals 180 degree
rightwards double arrow angle DAB plus angle BCD equals 180 degree
rightwards double arrow angle DAB equals 180 degree minus 110 degree space equals space 70 degree
Hence comma space correct space option space is space open parentheses straight a close parentheses. end style

Solution 5

begin mathsize 12px style Chord space AB space space equals space 14 space cm
AC space equals space BC equals 7 space cm
OC equals 6 space cm
rightwards double arrow OB space equals space square root of 7 squared space plus space 6 squared end root equals square root of 49 plus 36 end root equals square root of 85 cm
Consider space triangle ODE
open parentheses OE close parentheses squared plus open parentheses ED close parentheses squared space equals space left parenthesis OD right parenthesis squared
open parentheses 2 close parentheses squared space plus open parentheses ED close parentheses to the power of 2 space end exponent equals left parenthesis OD right parenthesis squared
open parentheses ED close parentheses to the power of 2 space end exponent equals open parentheses OD close parentheses to the power of 2 space end exponent minus 4
rightwards double arrow left parenthesis ED right parenthesis squared equals 85 minus 4 space space space space left parenthesis OD equals OB equals square root of 85 space cm space minus space radii space of space same space circle right parenthesis
rightwards double arrow ED space equals space square root of 81 space equals space 9 space cm
rightwards double arrow Chord space FD space equals space 9 space cross times space 2 space equals space 18 space cm
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 6

begin mathsize 12px style The space longest space chord space of space straight a space circle space is space its space diameter.
rightwards double arrow Diameter space greater than space 10 space cm
rightwards double arrow 2 space cross times space Radius space greater than 10 space cm
rightwards double arrow Radius space greater than space 5 space cm
Hence comma space correct space option space is space open parentheses straight b close parentheses.
end style

Solution 7

begin mathsize 12px style right enclose AD space and space AC space are space radii space of space same space circle space and space CD space is space straight a space chord.
Consider space triangle ABC comma
BC squared space equals open parentheses AC close parentheses squared minus open parentheses AB close parentheses squared space equals space 5 squared minus 4 squared space equals space 25 minus 16 space equals space 9
rightwards double arrow BC space equals space 3 space cm
Chord space CD space equals space 2 space cross times BC space equals space 6 space cm
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 8


begin mathsize 12px style Chord space AB space equals space Chord space BC space equals space Chord space CD
rightwards double arrow angle AOB space equals angle BOC equals angle COD space space space space space left parenthesis equal space chords space subtend space equal space angles space at space the space center right parenthesis
Now comma space angle AOB space plus space angle BOC plus angle COD equals 180 degree
rightwards double arrow angle AOB space plus space angle AOB plus angle AOB equals 180 degree
rightwards double arrow 3 angle AOB space equals 180 degree
rightwards double arrow angle AOB space equals space 60 degree
Hence comma space correct space option space is space open parentheses straight a close parentheses. end style

Solution 9

begin mathsize 12px style straight m space AB with overparenthesis on top equals 183 degree
straight O space is space the space centre space of space the space circle space and space AB space is space straight a space chord.
The space Regiion space bounded space by space arc space and space line space segment space AB space is space shaded.
We space can space see comma space apostrophe straight O apostrophe comma space the space centre comma space always space lie space in space the space interior space of space straight S.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 10

begin mathsize 12px style fraction numerator AB with overparenthesis on top minor over denominator AB with overparenthesis on top major end fraction equals 1 third equals fraction numerator angle space AB with overparenthesis on top minor over denominator angle space AB with overparenthesis on top major end fraction
Let space space angle AB with overparenthesis on top minor space equals space straight x
rightwards double arrow angle space stack AB space with overparenthesis on top major space equals space 3 straight x
Now space we space know space straight x plus 3 straight x equals 360 degree
rightwards double arrow 4 straight x equals 360 degree
rightwards double arrow straight x equals 90 degree
rightwards double arrow 3 straight x equals 270 degree
Hence comma space correct space option space is space open parentheses straight a close parentheses.

O p t i o n space left parenthesis c right parenthesis space c a n space a l s o space b e space a space p o s s i b i l i t y. end style

Solution 11

begin mathsize 12px style straight m open parentheses AB with overparenthesis on top close parentheses equals 260 degree
rightwards double arrow straight m open parentheses BA with overparenthesis on top close parentheses space equals space 100 degree
Now space Let space BA with overparenthesis on top space subtend space an space angle space straight theta space at space straight a space point space straight C space on space circle.
Now comma space we space know space that space angle space subtend space by space an space arc space at space the space centre space
is space double space the space angle space subtended space at space any space point space on space the space circle.
rightwards double arrow 100 degree equals 2 straight theta
rightwards double arrow straight theta equals 50 degree
Hence comma space correct space option space is space open parentheses straight c close parentheses. end style

Solution 12

begin mathsize 12px style angle BAC space equals space 60 degree space open parentheses angle space of space equilateral space triangle close parentheses
Arc space BC with overparenthesis on top space makes space angle space angle BAC space at space circle space and space angle BOC space at space centre space of space circle.
rightwards double arrow angle BAC space equals 1 half angle BOC
rightwards double arrow 2 space cross times space angle BAC equals angle BOC
rightwards double arrow 2 cross times 60 degree space equals space angle BOC
rightwards double arrow angle BOC space equals space 120 degree
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 13

begin mathsize 12px style AB space and space CD space are space diameters space of space straight a space circle space and space diameter space makes space 90 degree space at space any space point space on space circle.
rightwards double arrow angle CAD space equals space angle CBD space equals space angle BCA equals angle ADB space equals space 90 degree
Also comma space diagnols space AB space and space CD space are space perpendicular to space each space other.
Thus comma space ABCD space is space straight a space square.
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 14

begin mathsize 12px style ABC space is space an space arc space of space circle.
Take space point space straight D space in space the space alternative space segment space and space join space AD space and space CD.
angle ABC equals 135 degree space space space left parenthesis given right parenthesis
angle ABC plus angle ADC space equals space 180 degree space left parenthesis sum space of space opposite space angles space of space cyclic space quadrilateral space is space 180 degree right parenthesis
rightwards double arrow angle ADC equals 180 degree minus angle ABC space equals 180 degree space minus space 135 degree space equals space 45 degree
Now comma space angle AOC equals 2 space cross times space angle ADC space equals space 2 space cross times space 45 degree space equals space 90 degree
ABC with overparenthesis on top space equals space measure space of space the space central space angle space equals angle AOC space equals space 90 degree
rightwards double arrow Required space ratio space equals space fraction numerator arc space ABC with overparenthesis on top over denominator circumference space end fraction space equals space fraction numerator 90 degree over denominator 360 degree end fraction space equals space 1 fourth space equals space 1 space colon space 4
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.
end style

Solution 15

begin mathsize 12px style angle AOB space equals 60 degree space space left parenthesis Since space triangle AOB space is space equilateral space triangle right parenthesis
Now comma space angle ADB space equals space 30 degree space space space space space space
left parenthesis Since space chord space AB space makes space 60 degree space at space centre comma space same space chord space will space make space half space of space the space angle space
at space circumference space of space angle space made space at space centre right parenthesis
Now space angle ACB space is space angle space made space by space chord space at space minor space arc space of space circle.
ACBD space is space cyclic space Quadrilateral.
rightwards double arrow angle straight C space plus angle straight D space equals space 180 degree
rightwards double arrow angle ACB space plus angle ADB space equals space 180 degree
rightwards double arrow angle ACB space equals space 180 degree space minus space 30 degree space equals 150 degree
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.111

Solution 16

begin mathsize 12px style In space straight a space cyclic space quadrilateral comma space opposite space angles space are space supplementary.
rightwards double arrow angle straight P space plus angle straight R space equals 180 degree
Now comma space angle straight P space equals space 67 degree space plus space 72 degree space equals space 139 degree
Thus comma space angle straight R space equals space 180 degree space minus 139 degree equals 41 degree
straight i. straight e. space angle straight R space equals space angle QRS space equals 41 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 17

begin mathsize 12px style angle AOC space equals space angle AOB space plus space angle BOC space equals space 90 degree plus 120 degree space equals 210 degree
angle COA space equals space 360 degree minus 210 degree space equals space 150 degree
If space arc space COA with overparenthesis on top space makes space 150 degree space at space centre comma space then space it space will space make space half space of space angle space of space the space centre space
at space circumference.
rightwards double arrow angle CBA space or space angle ABC space equals fraction numerator 150 degree over denominator 2 end fraction equals 75 degree
Hence comma space correct space option space is space open parentheses straight b close parentheses. space end style

Solution 18

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

Hence, correct option is (c).

Solution 19

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

Hence, correct option is (b).

Solution 20

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).

Solution 21

begin mathsize 12px style angle CAB space equals space angle CDB space equals straight x degree space space space space space space.... left parenthesis Both space are space on space the space same space arc right parenthesis
Consider space triangle ODB comma
angle DOB space equals space 90 degree comma space angle OBD space equals space straight y comma space angle ODB space equals space straight x
In space triangle ODB comma
straight x plus space straight y plus space 90 degree space equals space 180 degree
rightwards double arrow straight x space plus space straight y space equals space 90 degree
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 22

begin mathsize 12px style angle AOC space is space made space by space arc space AC with overparenthesis on top space at space centre space and space angle ABC space is space made space by space AC with overparenthesis on top space on space circumference space
in space major space segment.
rightwards double arrow angle ABC space equals 1 half space angle AOC
rightwards double arrow angle AOC space equals space 2 space cross times space angle ABC space equals space 2 space cross times space 45 degree space equals space 90 degree
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 23

begin mathsize 12px style angle APC space plus angle APB space equals space 180 degree
rightwards double arrow angle APB space equals space 180 degree minus 90 degree equals 90 degree
In space triangle APB comma
angle ABP space equals 180 degree space minus space angle APB space minus space angle BAP space equals space 180 degree space minus space 90 degree minus 35 degree space equals space 55 degree
Now space Arc space AC with overparenthesis on top space makes space angle ABC space and space angle ADC space on space circle.
rightwards double arrow ABC equals angle ADC
rightwards double arrow angle ADC space equals space 55 degree
Hence comma space correct space option space is space open parentheses straight c close parentheses. end style

Solution 24

begin mathsize 12px style angle ABC space equals space 90 degree space space space space space.... open parentheses Diameter space AC space makes space 90 degree space at space circumference close parentheses
angle CDB equals space angle CAB space space space space space space.... open parentheses angles space on space the space same space arc close parentheses
rightwards double arrow angle CAB space equals space 42 degree
In space triangle ABC comma
angle ACB space equals 180 degree minus 90 degree minus space 42 degree equals 48 degree
Hence comma space correct space option space is space open parentheses straight b close parentheses. end style

RD Sharma Solution for Class 9 Mathematics Chapter 15 - Circles Page/Excercise 15.112

Solution 25

begin mathsize 12px style OC space equals space OA equals straight r space space space space left parenthesis radius right parenthesis
AB equals Diameter space equals space 2 straight r
AC equals square root of open parentheses OA close parentheses squared plus open parentheses OC close parentheses squared end root
space space space space space equals space square root of straight r to the power of space 2 end exponent plus straight r squared end root
space space space space space equals square root of 2 straight r
space space space space space equals space square root of 2 open parentheses AB over 2 close parentheses
rightwards double arrow AC space equals space fraction numerator 1 over denominator square root of 2 end fraction AB
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 26

begin mathsize 12px style Both space the space circles space pass space through space the space center space of space each space other
rightwards double arrow straight O subscript 1 straight O subscript 2 equals straight r
Common space chord space is space AB.
We space know space that space perpendicular space drawn space from space centre space of space circle space to space any space chord space bisects space it.
rightwards double arrow straight P space is space the space midpoint space of space AB
rightwards double arrow PA equals PB
straight O subscript 1 straight A equals straight r space space space space left parenthesis radius space of space circle right parenthesis
Consider space triangle straight O subscript 1 PA
open parentheses straight O subscript 1 straight A close parentheses squared equals AP squared space plus space straight O subscript 1 straight P squared space space
rightwards double arrow straight r squared equals AP squared space plus open parentheses straight r over 2 close parentheses squared space space space space space space space.... open parentheses straight P space is space also space mid minus poinat space of space straight O subscript 1 straight O subscript 2 space space end subscript close parentheses
rightwards double arrow AP squared space equals space straight r squared minus straight r squared over 4 equals fraction numerator 3 straight r squared over denominator 4 end fraction
rightwards double arrow AP space equals space fraction numerator square root of 3 over denominator 2 end fraction straight r
rightwards double arrow Length space of space chord space AB equals 2 AP equals square root of 3 straight r
Hence comma space correct space option space is space open parentheses straight c close parentheses. end style

Solution 27

begin mathsize 12px style angle APB space and space angle AQB space are space on space the space same space arc.
rightwards double arrow angle APB space equals space angle AQB
But comma space if space AB equals diameter comma space then space angle APB space equals space angle AQB space equals space 90 degree
open parentheses Becasue space diameter space makes space Right space angle space at space any space point space on space circumference space of space circle close parentheses
rightwards double arrow angle APB space plus angle AQB space equals space 180 degree
Hence comma space correct space option space is space open parentheses straight b close parentheses. end style

Solution 28

begin mathsize 12px style OB space and space OD space are space the space radii space of space straight a space circle.
In space triangle OED comma
straight r squared equals OE squared space plus ED squared equals OE squared space plus space open parentheses 6 close parentheses squared
rightwards double arrow OE space equals space square root of straight r squared space minus space 36 space end root space space space space space space space space space space space space space space space.... open parentheses 1 close parentheses
In space triangle OFB comma
straight r squared space equals OF squared plus BF squared equals OF squared space plus open parentheses 3 close parentheses squared
rightwards double arrow OF space equals space square root of straight r squared space minus space 9 end root space space space space space space space space space space space space.... open parentheses 2 close parentheses
OF space minus space OE space equals space 3 space cm space space space space space space space open parentheses given close parentheses
rightwards double arrow square root of straight r squared minus 9 end root space minus square root of straight r squared minus 36 end root space equals space 3 space
rightwards double arrow square root of straight r squared minus 9 end root equals square root of straight r squared minus 36 end root plus 3 space space space space.... open parentheses 3 close parentheses
Squarring space equation space open parentheses 3 close parentheses comma space we space have
straight r squared space minus 9 space equals space straight r squared space minus space 36 space plus space 9 space plus space 2 cross times 3 space square root of straight r squared minus 36 end root
rightwards double arrow straight r squared space minus space 9 space equals space straight r squared space minus space 27 space plus space 6 space square root of straight r squared minus 36 end root
rightwards double arrow 18 space equals space 6 space square root of straight r squared minus 36 end root
rightwards double arrow 3 space equals space square root of straight r squared minus 36 end root
rightwards double arrow 9 space equals space straight r squared minus 36 space
rightwards double arrow straight r equals square root of 45 space equals space 3 square root of 5 space cm
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 29

begin mathsize 12px style PQ space equals space 23 space cm
AB space equals space 16 space cm space
rightwards double arrow BP space equals space AP equals space 8 space cm
straight r space equals space 17 space cm
rightwards double arrow EF space equals space diameter space equals space 2 straight r space equals space 34 space cm
Consider space triangle OPB comma
straight r squared equals OP squared plus BP squared
rightwards double arrow OP squared equals open parentheses 17 close parentheses squared minus open parentheses 8 close parentheses squared space equals 289 space minus space 64 space equals space 225
rightwards double arrow OP space equals space 15 space cm
rightwards double arrow OQ space equals space 23 minus 15 space equals space 8 space cm
Consider space triangle OQD comma
straight r squared space equals space OQ squared plus QD squared
rightwards double arrow QD squared space space equals space straight r squared minus OQ squared equals open parentheses 17 close parentheses squared minus open parentheses 8 close parentheses squared space equals space 225
rightwards double arrow QD space equals space 15 space cm
rightwards double arrow CD equals space 2 cross times space QD space equals space 30 space cm
Hence comma space correct space option space is space open parentheses straight d close parentheses. end style

Solution 30

begin mathsize 12px style angle ADC space equals 1 half angle AOC
open curly brackets angle ADC space and space angle AOC space are space made space by space same space AC with overparenthesis on top space on space centre space and space circumference close curly brackets
rightwards double arrow angle ADC equals 1 half cross times 130 degree space equals space 65 degree
ADCB space is space straight a space cyclic space Quarilateral. space
rightwards double arrow angle straight D plus angle straight B equals 180 degree
rightwards double arrow angle ABC space equals space 180 degree space minus 65 degree space equals space 115 degree
Hence comma space correct space option space is space open parentheses straight b close parentheses. end style

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 15 - Circles for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 15 - Circles.

Text Book Solutions

CBSE IX - Mathematics

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