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

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

Edited By Ramraj Saini | Updated on Apr 23, 2024 12:10 PM IST

NCERT Solutions for Class 9 Maths Exercise 10.4 Chapter 10 Circles- Download Free PDF

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.

The 9th class maths exercise 10.4 answers consists of six questions expertly developed by Careers360 subject matter specialists. These class 9 maths chapter 10 exercise 10.4 provide students with complete help by providing extensive, step-by-step explanations. PDF versions are also freely accessible for download, increasing accessibility and convenience.

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

**Please be aware that this chapter has been renumbered as Chapter 9 in the CBSE Syllabus for the academic year 2024-25.

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NCERT Solutions for Class 9 Maths Chapter 10 – Circles Exercise 10.4

Q1 Two circles of radii \small 5\hspace{1mm}cm and \small 3\hspace{1mm}cm intersect at two points and the distance between their centres is \small 4\hspace{1mm}cm . Find the length of the common chord.
Answer:

Given: Two circles of radii \small 5\hspace{1mm}cm and \small 3\hspace{1mm}cm intersect at two points and the distance between their centres is \small 4\hspace{1mm}cm .

To find the length of the common chord.

Construction: Join OP and draw OM\perp AB\, \, and \, \, \, ON\perp CD.

1640237228523

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

\therefore PM\perp AB

\angle PMA=90 \degree

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

In \triangle APM, using Pythagoras theorem

AM^2=AP^2-PM^2 ...........................1

Also,

In \triangle AQM, using Pythagoras theorem

AM^2=AQ^2-MQ^2 ...........................2

From 1 and 2, we get

AP^2-PM^2=AQ^2-MQ^2

\Rightarrow 3^2-x^2=5^2-(4-x)^2

\Rightarrow 9-x^2=25-16-x^2+8x

\Rightarrow 9=9+8x

\Rightarrow 8x=0

\Rightarrow x=0

Put,x=0 in equation 1

AM^2=3^2-0^2=9

\Rightarrow AM=3

\Rightarrow AB=2AM=6

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.
Answer:

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 OM\perp AB\, \, \, \, and\, \, \, ON\perp CD.

Proof :

1640237267424

In \triangle OMP and \triangle ONP,

AP = AP (Common)

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

\angle OMP = \angle ONP (Both are right angled)

Thus, \triangle OMP \cong \triangle ONP (By SAS rule)

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

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

\Rightarrow \frac{1}{2}AB=\frac{1}{2}CD

\Rightarrow AM = CN ......................3

Adding 1 and 3, we have

AM + PM = CN + PN

\Rightarrow AP = CP

Subtract 4 from 2, we get

AB-AP = CD - CP

\Rightarrow PB = PD

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.
Answer:

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. \angle OPM= \angle OPN

Proof :

Construction: Join OP and draw OM\perp AB\, \, \, \, and\, \, \, ON\perp CD.

In \triangle OMP and \triangle ONP,

AP = AP (Common)

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

\angle OMP = \angle ONP (Both are right-angled)

Thus, \triangle OMP \cong \triangle ONP (By RHS rule)

\angle OPM= \angle 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 \small AB=CD (see Fig. \small 10.25 ).

Answer:

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 OM\perp AD

Proof :

1640237314606

BC is a chord of the inner circle and OM\perp BC

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

(Perpendicular OM bisect BC)

Similarly,

AD is a chord of the outer circle and OM\perp AD

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

(Perpendicular OM bisect AD )

Subtracting 1 from 2, we get

AM-BM = DM - CM

\Rightarrow AB = CD


Q5 Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius \small 5m 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 \small 6m each, what is the distance between Reshma and Mandip?

Answer:

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 OL\perp RS .

Proof:

1640237339300 In \triangle ORS,

OS = OR and OL\perp RS (by construction )

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

In \triangle OLS, by pytagoras theorem,

OL^2=OS^2-SL^2

\Rightarrow OL^2=5^2-3^2=25-9=16

\Rightarrow OL=4

In \triangle ORK and \triangle OMK,

OR = OM (Radii)

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

OK = OK (Common)

\triangle ORK \cong \triangle OMK (By SAS)

RK = MK (CPCT)

Thus, OK\perp RM

area of \triangle ORS = \frac{1}{2}\times RS\times OL ...............................1

area of \triangle ORS = \frac{1}{2}\times OS\times KR .............................2

From 1 and 2, we get

\frac{1}{2}\times RS\times OL =\frac{1}{2}\times OS\times KR

\Rightarrow RS\times OL=OS\times KR

\Rightarrow 6\times 4=5\times KR

\Rightarrow KR=4.8 cm

Thus, RM =2 KR=2\times 4.8 cm=9.6 cm

Q6 A circular park of radius \small 20m 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.

Answer:

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 \perp SD

Proof :

1640237370579

Let AS = SD = AD = 2x cm

In \triangle ASD,

AS = AD and AP \perp SD

So, SP = PD = x cm

In \triangle OPD, by Pythagoras,

OP^2=OD^2-PD^2

\Rightarrow OP^2=20^2-x^2=400-x^2

\Rightarrow OP=\sqrt{400-x^2}

In \triangle APD, by Pythagoras,

AP^2=AD^2-PD^2

\Rightarrow (AO+OP)^2+x^2=(2x)^2

\Rightarrow (20+\sqrt{400-x^2})^2+x^2=4x^2

\Rightarrow 400+400-x^2+40\sqrt{400-x^2}+x^2=4x^2

\Rightarrow 800+40\sqrt{400-x^2}=4x^2

\Rightarrow 200+10\sqrt{400-x^2}=x^2

\Rightarrow 10\sqrt{400-x^2}=x^2-200

Squaring both sides,

\Rightarrow 100(400-x^2)=(x^2-200)^2

\Rightarrow 40000-100x^2=x^4-40000-400x^2

\Rightarrow x^4-300x^2=0

\Rightarrow x^2(x^2-300)=0

\Rightarrow x^2=300

\Rightarrow x=10\sqrt{3}

Hence, length of string of each phone = 2x=20\sqrt{3} m

More About NCERT Solutions for Class 9 Maths Exercise 10.4

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.'
  • 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.

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

Key Features of Exercise 10.4 Class 9 Maths

  • 9th class maths exercise 10.4 answers is a comprehensive exercise that covers various concepts related to circles, including chords, secants, and tangents.
  • Class 9 maths ex 10.4 offers a variety of problems with different levels of complexity. This allows students to practice and strengthen their understanding of circle-related concepts.
  • Solutions to the problems in class 9 ex 10.4 are presented in a step-by-step format. This helps students follow the solution process and understand the concepts involved.
  • Solutions for class 9 maths ex 10.4 are often available in PDF format. This enables students to download and access them offline for convenient studying.
  • Exercise 10.4 aligns with the prescribed syllabus, covering all relevant topics and concepts related to circles in Class 9 Maths.

Also, See

NCERT Solutions of Class 10 Subject Wise

Subject Wise NCERT Exemplar Solutions

Frequently Asked Question (FAQs)

1. What is the main concept of NCERT solutions for Class 9 Maths exercise 10.4?

In this exercise, we learn about the chords and their related theorems such as Chords that are equidistant from the circle's centre have the same length.

2. What is the angle subtended by the largest chord of the circle on circumference?

The largest chord subtends the right angle on the circumference.

3. The number of equal chords in the circle is finite?

NO, because equal chords can be infinite.

4. If the chord is equal to the radius of the circle, what angle will it subtend at a point on the major arc?

The subtended angle of the chord at a position on the major arc is 30. An equilateral triangle has 60 degrees on each side. An arc of a circle at its centre has double the angle of any other point of the circle.

5. Every chord is also the diameter of a circle. Is it also true that the converse of this statement is true?

Because its endpoints are on the circle's circumference, every diameter is a chord. It's the longest chord that runs through the circle's centre.

However, as every chord does not pass through the centre, every chord cannot be a diameter.

6. What is the definition of con cyclic points?

Concyclic points are a group of points that all lie in the same circle.

7. When a circle is divided in three equal arcs, each arc is major arc. Is it true or false?

False, because a major arc's measure is more than 180° and equal to 360° minus the minor arc's measure with the same endpoints.

8. The length of perpendicular OE on CD = 5 cm when two equal chords AB and CD of a circle. When OF is perpendicular on AB. What is the length of OF?

The length of OF is also 5cm because Chords that are equidistant from the circle's centre have the same length.

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Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
Choreographer

The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

2 Jobs Available
Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available
Producer

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story. 

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

2 Jobs Available
Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook. 

5 Jobs Available
Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

3 Jobs Available
Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.  

2 Jobs Available
Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications. 

2 Jobs Available
Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

5 Jobs Available
QA Manager
4 Jobs Available
Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
Production Manager
3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
QA Lead

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans. 

2 Jobs Available
Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software. 

2 Jobs Available
Process Development Engineer

The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.

2 Jobs Available
QA Manager
4 Jobs Available
AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

4 Jobs Available
Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

4 Jobs Available
Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

3 Jobs Available
ITSM Manager
3 Jobs Available
Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process. 

2 Jobs Available
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