RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

RD Sharma Solutions for Class 9 Mathematics

Created by: Abhishek Kapoor

Class 9 : RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

The document RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.
All you need of Class 9 at this link: Class 9

Q1) The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.

Solution:

Given that,

Radius of circle (OA) = 8cm

Chord (AB) = 12cm

Draw OC⊥AB

We know that

The perpendicular from centre to chord bisects the chord

∴ AC = BC = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = 6cm

Now in ΔOCA, by Pythagoras theorem

AC+ OC = OA2

⇒ 62 + OC2  = 82

⇒ 36 + OC2  = 64

⇒ OC2  = 64-36

⇒ OC2  = 28

⇒ OC =  RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

⇒ OC = 5.291cm

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

 

Q2) Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Given that,

Distance (OC) = 5cm

Radius of the circle (OA) = 10cm

In ΔOCA, by Pythagoras theorem

OC+ AC = OA2

⇒ 52 + AC2  = 102

⇒ 25 + AC2  = 100

⇒ AC2  = 100 – 25

⇒ AC2  = 75

⇒ AC = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

⇒ AC = 8.66cm

We know that, the perpendicular from the centre to chord bisects the chord

Therefore, AC = BC = 8.66cm

Then the chord AB = 8.66 + 8.66

= 17.32cm

 

Q3) Find the length of a chord which is at a distance of 4 cm from the centre of a circle of radius 6 cm.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Given that,

Radius of the circle (OA) = 6cm

Distance (OC) = 4cm

In ΔOCA, by Pythagoras theorem

AC+ OC = OA2

⇒ AC2 + 42  = 62

⇒ AC2  = 36 – 16

⇒ AC2  = 20

⇒ AC =  RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

⇒ AC = 4.47cm

We know that the perpendicular distance from centre to chord bisects the chord.

AC = BC = 4.47cm

Then AB = 4.47 + 4.47

= 8.94cm

 

Q4) Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel. If the distance between AB and CD is 3 cm, find the radius of the circle.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Construction: Draw OP⊥CD

Chord AB = 5cm

Chord CD = 11cm

Distance PQ = 3cm

Let OP = x cm

And OC = OA = r cm

We know that the perpendicular from centre to chord bisects it.

∴ CP = PD = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRevcm

And AQ = BQ = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRevcm

In ΔOCP, by Pythagoras theorem

OC2 = OP2+CP2

⇒r2 = x2+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev…..(i)

In ΔOQA, by Pythagoras theorem

OA2 = OQ2+AQ2

⇒r2 = (x+3)2+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev…..(ii)

Compare equation (i) and (ii)

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

⇒x2−x2+6x = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev−9

⇒6x = 15

⇒x = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

 

Q5) Give a method to find the centre of a given circle.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Steps of Construction:

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

(2) Join AB and BC.

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

(4) Point O will give the required circle because we know that, the Perpendicular bisectors of chord always pass through the centre.

 

Q6) Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Given:

C is the mid-point of chord AB.

To prove: D is the mid-point of arc AB.

Proof:

In ΔOACandΔOBC

OA = OB [Radius of circle]

OC = OC [Common]

AC = BC [C is the mid-point of AB]

ThenΔOAC≅ΔOBC[BySSScondition]

∴ ∠AOC = ∠BOC

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev
RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Hence, D is the mid-point of arc AB.

 

Q7) Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Given:

PQ is a diameter of circle which bisects the chord AB at C.

To Prove: PQ bisects ∠AOB

Proof:

In ∠AOCand∠BOC

OA = OB [Radius of circle]

OC = OC [Common]

AC = BC [Given]

Then ΔAOC≅ΔBOC          [By SSS  condition]

∠AOC = ∠BOC[C.P.C.T]

Hence PQ bisects ∠AOB.

 

Q8) Prove that two different circles cannot intersect each other at more than two points.

Solution:

Suppose two circles intersect in three points A, B, C.

Then A, B, C are non-collinear so a unique circle passes through these three points. This is contradiction to the face that two given circles are passing through A, B, C. Hence, two circles cannot intersect each other at more than two points.

 

Q9) A line segment AB is of length 5 cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your answer.

Solution:

(1) Draw a line segment AB of 5cm.

(2) Draw the perpendicular bisectors of AB.

(3) With centre A and radius of 4cm, draw an arc which intersects the perpendicular bisector at point O. The point O will be the required centre.

(4) Join OA.

(5) With centre O and radius OA, draw a circle.

No, we cannot draw a circle of radius 2cm passing through A and B because when we draw an arc of radius 2cm with centre A, the arc will not intersect the perpendicular bisector and we will not find the centre.

Q10) An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Let ABC be an equilateral triangle of side 9cm and let AD is one of its median.

Let G be the centroid of ΔABC. Then AG : GD = 2 : 1

We know that in an equilateral triangle, centroid coincides with the circum centre.

Therefore, G is the centre of the circumference with circum radius GA.

Also G is the centre and GD is perpendicular to BC.

Therefore, In right triangle ADB, we have

AB2  = AD2 + DB2

⇒ 92  = AD2 + DB2

⇒AD =  RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

∴ Radius = AG = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

 

Q11) Given an arc of a circle, complete the circle.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Steps of Construction:

(1) Take three points A, B and C on the given arc.

(2) Join AB and BC.

(3) Draw the perpendicular bisectors of chords AB and BC which intersect each other at point O. Then O will be the required centre of the required circle.

(4) Join OA.

(5) With centre O and radius OA, complete the circle.

 

Q12) Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Solution:

Each pair of circles have 0, 1 or 2 points in common.

The maximum number of points in common is ‘2’.

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

 

Q13) Suppose you are given a circle. Give a construction to find its centre.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Steps of Construction:

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

(2) Join AB and BC.

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

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

 

Q14) Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are opposite side of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Draw OM⊥ABandON⊥CD.

Join OB and OD.

BM = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev[Perpendicular from the centre bisects the chord]

ND = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

Let ON be x, so OM will be 6 – x.

ΔMOB

OM2 + MB2  = OB2

(6−x)2+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = OB2

36+x2−12x+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = OB2…..(i)

In ΔNOD

ON2 + ND2  = OD2

x2+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = OD2

x2+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = OD2…..(ii)

We have OB = OD. [Radii of same circle]

So, from equation (i) and (ii).

36+x2−12x+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

⇒12x = 36+RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

x = 1

From equation (ii)

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

OD2 = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

OD = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

So, the radius of the circle is found to be 55√2cm

 

Q15) The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?

Solution:

RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

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

MB = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = 3cm

InΔOMB

OM2+MB2 = OB2

42+92 = OB2

16+9 = OB2

OB = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

OB = 5cm

InΔOND

OD = OB = 5cm              [Radii of same circle]

ND = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev = 4cm

ON2+ND2 = OD2

ON2+42 = 52

ON2 = 25−16

ON = RD Sharma Solutions -Ex-16.2, Circles, Class 9, Maths Class 9 Notes | EduRev

ON = 3cm

So, the distance of bigger chord from the circle is 3cm.

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