# NCERT Solutions for Exercise 10.4 Class 9 Maths Chapter 10 - Circles

NCERT Solutions for Class 9 Maths exercise 10.4 - We'll start with a basic definition of a circle and then go over the theorems discussed in chapter 10 of Class 9 Math exercise 10.4. Any closed shape with all points connected at the same distance from the centre is called a circle. Any point equidistant from any of the circle's boundaries is the centre of the circle. Radius is a Latin word that means 'ray,' but it refers to the line segment that connects the circle's centre and edge. Any line that begins or ends at the circle's centre and connects to any point on the circle's border is defined as the radius of the circle.

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NCERT solutions Class 9 Maths exercise 10.4 – This exercise includes some important theorems about two equal chords for the examination point of view. We'll start with a definition of a chord.

Chord: A chord is a straight line segment with both ends on the circle's perimeter. Its Latin translation is 'bowstring.'

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The length of the perpendicular from a point to a line defines the distance between them.

The theorems are:

A circle's equal chords (or congruent circles' equal chords) are equidistant from the centre (or centres).

Chords that are equidistant from the circle's centre have the same length.

Along with NCERT book Class 9 Maths chapter 9 exercise 10.4 the following exercises are also present.

• Circles Exercise 10.1

• Circles Exercise 10.2

• Circles Exercise 10.3

• Circles Exercise 10.5

• Circles Exercise 10.6

## Circles Class 9 Chapter 10 Exercise: 10.4

Q1 Two circles of radii and intersect at two points and the distance between their centres is . Find the length of the common chord.

Given: Two circles of radii and intersect at two points and the distance between their centres is .

To find the length of the common chord.

Construction: Join OP and draw

Proof: AB is a chord of circle C(P,3) and PM is the bisector of chord AB.

Let, PM = x , so QM=4-x

In APM, using Pythagoras theorem

...........................1

Also,

In AQM, using Pythagoras theorem

...........................2

From 1 and 2, we get

Put,x=0 in equation 1

Q2 If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Given: two equal chords of a circle intersect within the circle

To prove: Segments of one chord are equal to corresponding segments of the other chord i.e. AP = CP and BP=DP.

Construction : Join OP and draw

Proof :

In OMP and ONP,

AP = AP (Common)

OM = ON (Equal chords of a circle are equidistant from the centre)

OMP = ONP (Both are right angled)

Thus, OMP ONP (By SAS rule)

PM = PN..........................1 (CPCT)

AB = CD ............................2(Given )

......................3

Adding 1 and 3, we have

AM + PM = CN + PN

Subtract 4 from 2, we get

AB-AP = CD - CP

Q3 If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Given: two equal chords of a circle intersect within the circle.

To prove: the line joining the point of intersection to the centre makes equal angles with the chords.
i.e. OPM= OPN

Proof :

Construction: Join OP and draw

In OMP and ONP,

AP = AP (Common)

OM = ON (Equal chords of a circle are equidistant from the centre)

OMP = ONP (Both are right-angled)

Thus, OMP ONP (By RHS rule)

OPM= OPN (CPCT)

Q4 If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that (see Fig. ).

Given: a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D.

To prove : AB = CD

Construction: Draw

Proof :

BC is a chord of the inner circle and

So, BM = CM .................1

(Perpendicular OM bisect BC)

Similarly,

AD is a chord of the outer circle and

So, AM = DM .................2

Subtracting 1 from 2, we get

AM-BM = DM - CM

Q5 Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is each, what is the distance between Reshma and Mandip?

Given: From the figure, R, S, M are the position of Reshma, Salma, Mandip respectively.

So, RS = SM = 6 cm

Construction : Join OR,OS,RS,RM and OM.Draw .

Proof:

In ORS,

OS = OR and (by construction )

So, RL = LS = 3cm (RS = 6 cm )

In OLS, by pytagoras theorem,

In ORK and OMK,

ROK = MOK (Equal chords subtend equal angle at centre)

OK = OK (Common)

ORK OMK (By SAS)

RK = MK (CPCT)

Thus,

area of ORS = ...............................1

area of ORS = .............................2

From 1 and 2, we get

Thus,

Q6 A circular park of radius is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

Given: In the figure, A, S, D are positioned Ankur, Syed and David respectively.

So, AS = SD = AD

Radius of circular park = 20 m

so, AO=SO=DO=20 m

Construction: AP SD

Proof :

Let AS = SD = AD = 2x cm

In ASD,

AS = AD and AP SD

So, SP = PD = x cm

In OPD, by Pythagoras,

In APD, by Pythagoras,

Squaring both sides,

Hence, length of string of each phone m

## More About NCERT Solutions for Class 9 Maths Exercise 10.4

The diameter is the longest chord, and all diameters are the same length: two times the radius.

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An arc is a circle segment that connects two points. In a circle, equal chords have equal arcs.

A chord that passes through the centre is referred to as a diameter.

The circumference of a circle is the measurement of its circumference.

A segment of the circle is the area between a chord and one of its arcs.

A sector is an area between an arc and the two radii that connect the arc's centre and endpoints.

Also Read| Circles Class 9 Notes

## Benefits of NCERT Solutions for Class 9 Maths Exercise 10.4

• Equal Chords and Their Distances from the Center is the subject of exercise 10.4 in Class 9 Math.

• NCERT syllabus Class 9 Maths chapter 10 exercise 10.4 introduces us to theorems related to the equal chords in a circle.

• Understanding the principles from chapter 10 exercise 10.4 in Class 9 Arithmetic will help us grasp the theorems of equal chords and their distance.

Also, See

• NCERT Solutions for Class 9 Maths Chapter 10

• NCERT Exemplar Solutions Class 9 Maths Chapter 10

## NCERT Solutions of Class 10 Subject Wise

• NCERT Solutions for Class 9 Maths

• NCERT Solutions for Class 9 Science

## Subject Wise NCERT Exemplar Solutions

• NCERT Exemplar Class 9 Maths
• NCERT Exemplar Class 9 Science