NCERT Solutions for Class 9 Maths Chapter 11 Constructions

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# NCERT Solutions for Class 9 Maths Chapter 11 Constructions

Edited By Ramraj Saini | Updated on May 11, 2023 03:49 PM IST

## NCERT Solutions for Class 9 Maths Chapter 11 Constructions

NCERT solutions for class 9 maths chapter 11 Constructions are discusses in this article. These NCERT solutions are prepared by expert team at Careers360 considering the latest CBSE syllabus 2023-24. These solutions are designed in simple, easy to understand and comprehensive way to help students. The chapter construction class 9 deals with the construction of certain angles and also triangles in certain conditions.

When there is any construction project to start, then the first step to initiate that project is the planning and drawing. Geometrical constructions have their own line of study. If you want to pursue your career in this domain then the higher study related to this can be civil engineering and architecture. In this construction class 9 NCERT book chapter, there are 2 exercises having a total of 14 questions. Constructions Class 9 Questions And Answers have an in-depth solution to each and every question present in the practice exercises. For 360-degree help in your school examination preparation, you can use NCERT solutions for class 9 Maths.

## Constructions Class 9 Solutions - Important Points

Construction of Bisector of a Line Segment:

• This construction task involves finding the midpoint of a given line segment. To construct the bisector, draw two congruent arcs from each endpoint of the segment. Where these arcs intersect is the midpoint of the line segment.

Construction of Bisector of a Given Angle:

• Place the compass at the vertex of the angle.

• Draw an arc that intersects both arms of the angle.

• Repeat the process from the other arm of the angle.

• The point where the two arcs intersect is the angle bisector.

Construction of an Equilateral Triangle:

• Draw a line segment to represent one side of the equilateral triangle.

• Place the compass at one endpoint of the line segment and draw an arc.

• Repeat the process from the other endpoint.

• Where the two arcs intersect, connect it to the endpoints of the line segment. This forms an equilateral triangle.

Construction of a Triangle Given Its Base, Sum of the Other Two Sides, and One Base Angle:

• This construction involves constructing a triangle when you know one of its base angles, the length of the base, and the sum of the lengths of the other two sides. Specific construction steps would depend on the given measurements.

Construction of a Triangle Given Its Base, Difference of the Other Two Sides, and One Base Angle:

• Similar to the previous construction, this involves constructing a triangle when you know one of its base angles, the length of the base, and the difference between the lengths of the other two sides. Specific construction steps would depend on the given measurements.

Construction of a Triangle of Given Perimeter and Two Base Angles:

• This construction task involves constructing a triangle with specified base angles and a given perimeter. Specific construction steps would depend on the given measurements.

Free download NCERT Solutions for Class 9 Maths Chapter 11 Constructions for CBSE Exam.

## Constructions Class 9 NCERT Solutions (Intext Questions and Exercise)

Class 9 maths chapter 11 question answer - Exercise: 11.1

The steps of construction to follow:

Step 1: Draw a ray OP.

Then, take O as the centre and any radius draw an arc cutting OP at Q.

Step 2: Now, taking Q as the centre and with the same radius as before draw an arc cutting the previous arc at R. Repeat the process with R to cut the previous arc at S.

Step 3: Take R and S as centre draw the arc of radius more than the half of RS and draw two arcs intersecting at A. Then, join OA.

Hence, $\angle POA = 90^{\circ }$ .

Justification:

We need to justify, $\angle POA = 90^{\circ }$

So, join OR and OS and RQ. we obtain

By construction OQ = OS = QR.

So, $\triangle ROQ$ is an equilateral triangle. Similarly $\triangle SOR$ is an equilateral triangle.

So, $\angle SOR = 60^{\circ}$

Now, $\angle ROQ = 60^{\circ}$ that means $\angle ROP = 60^{\circ}$ .

Then, join AS and AR:

Now, in triangles OSA and ORA:

$SR = SR$ (common)

$AS = AR$ (Radii of same arcs)

$OS = OR$ (radii of the same arcs)

So, $\angle SOA = \angle ROA = \frac{1}{2}\angle SOR$

Therefore, $\angle ROA = 30^{\circ}$

and $\angle POA = \angle ROA+\angle POR = 30^{\circ} +60^{\circ} =90^{\circ}$

Hence, justified.

The steps of construction to follow:

Step 1: Draw a ray OY.

Then, take O as the centre and any radius, mark a point A on the arc ABC.

Step 2: Now, taking A as the centre and the same radius, mark a point B on the arc ABC.

Step 3: Take B as a centre and the same radius, mark a point C on the arc ABC.

Step 4: Now, taking C and B as centre one by one, draw an arc from each centre intersecting each other at a point X.

Step 5: X and O are joined and a ray making an angle $90^{\circ}$ with OY is formed.

Let the arc AC touches OX at E

Step 6: With A and E as centres, 2 arcs are marked intersecting each other at D and the bisector of angle XOY is drawn.

Justification:

By construction we have,

$\angle XOY = 90^{\circ}$

We constructed the bisector of $\angle XOY$ as $\angle DOY$

Thus,

$\angle DOY = \frac{1}{2}\angle XOY = \frac{1}{2}\times90^{\circ} = 45^{\circ}$

Steps to construction to follow:

Step 1: Draw a ray OY.

Step 2: Now, take A as a center and take any radius, then draw an arc AB cutting OY at A.

Step 3: Take A and B as centres, draw 2 arcs are marked intersecting each other at X and hence, the bisector of $30^{\circ}$ is constructed.

Thus, $\angle XOY$ is the required angle.

Steps to construction to follow:

Step 1: Draw a ray OY.

Step 2: Now, take A as a centre and take any radius, then mark a point B on the arc

Step 3: Take B as a centre with the same radius, mark a point C on the arc ABC.

Step 4: Now, taking B and C as centres simultaneously, then draw an arc from each centre intersecting each other at a point X. Then join X and O and a ray making an angle with OY is formed.

Let the arc AC touches OX at E

Step 5: Now, with A and E as centres, mark 2 arcs which intersect each other at D and we obtain the bisector of the angle $XOY$ .

Steps to construction to follow:

Step 1: Draw a ray OY.

Step 2: Now, take A as a centre and take any radius, draw an arc AB which cuts OY at A.

Step 3: Take A and B as a centre, then mark 2 arcs which intersect each other at X and Hence, the bisector is constructed of $30^{\circ}$ .

Step 4: Now, with A and E as centres, mark 2 arcs which intersect each other at D and we obtain the bisector of the angle $XOY$ .

Thus, the angle of $15^{\circ}$ is obtained which is $\angle EOY.$

Steps to construction to follow:

Step 1: Draw a ray OY.

Step 2: Now, taking O as the centre draw an arc ABC.

Step 3: On taking A as a centre, draw two arcs B and C on the arc ABC.

Step 4: Now, taking B and C as centres, arcs are made to intersect at point E and the $\angle EOY = 90^{\circ}$ is constructed.

Step 5: Taking A and C as centres, arcs are made to intersect at D.

Step 6: Now, join OD and hence, $\angle DOY = 75^{\circ}$ is constructed.

Hence, $\angle DOY$ is the required angle.

The steps of construction to be followed:

Step 1: Draw a ray OY.

Step 2: Then, taking O as a centre, draw an arc ABC.

Step 3: Now, with A as a centre, draw two arcs B and C which are made on the arc ABC.

Step 4: Taking B and C as centres simultaneously, arcs are made to intersect at E and $\angle EOY =90^{\circ}$ is constructed.

Step 5: With B and C as centres, arcs are made to intersect at X.

Step 6: Join the OX and we get $\angle XOY =105^{\circ}$ is constructed.

Thus, the angle $XOY$ is $105^{\circ}.$

The steps of construction to be followed:

Step 1: Draw a ray DY.

Step 2: Draw an arc ACD with O as a center.

Step 3: Now, with A as a centre, draw two arcs B and C on the arc ACD.

Step 4: Taking B and C as centres, arcs are made to intersect at E and the angle formed is $\angle EOY = 90^{\circ}$ .

Step 5: Take F and D as centres, draw arcs to intersect at point X or the bisector of angle EOD is made.

Step 6: Join OX and the $\angle XOY = 135^{\circ}$ is made.

Hence, the angle required $\angle XOY$ is $135^{\circ}$ .

The following steps to make an equilateral triangle:

Step 1: Draw a line segment AB = 4 cm.

Step 2: With A and B as centres, make two arcs in the line segment AB. Mark it as D and E respectively.

Step 3: Now, with D and E as centres, make the two arcs cutting the previous arcs respectively, and forming an angle of $60^{\circ }$ each.

Step 4: Extend the lines of A and B until they intersect each other at point C.

Hence, triangle constructed is ABC which is equilateral.

Now, Justification ;

Since the angles constructed are of $60^{\circ }$ each, so the third angle will also be $60^{\circ }$ .

$\left [ \because The\ sum\ of\ all\ angles\ of\ triangle = 180^{\circ} \right ]$

Class 9 maths chapter 11 NCERT solutions - Excercise: 11.2

The steps of construction are as follows:

Step 1: Draw a line segment BC of 7cm length. Taking the help of protractor make an $\angle XBC = 75^{\circ}$ .

Step 2: Now, cut a line segment BD having 13 cm on BX which is $(AB+AC).$

Step 3: Now, join CD.

Step 4: Draw a perpendicular bisector of CD to intersect BD at a point A. Join AC. Then ABC is the required triangle

Hence, the required triangle is ABC.

The steps of construction to be followed:

Step 1: Draw a line segment BC = 8cm and make an angle of $45^{\circ}$ at point B i.e., $\angle XBC.$

Step 2: Now, cut the line segment BD = 3.5 cm on ray BX. i.e. $(AB-AC)$ .

Step 3: Join CD and draw a perpendicular bisector of CD i.e., PQ.

Step 4: Let the perpendicular bisector of CD intersects BX at point A. Then,

Step 5: Join AC, to get the required triangle $\triangle ABC.$

The steps of construction to be followed:

Step 1: Draw a ray QX and cut off a line segment QR which is equal to 6 cm in length.

Step 2: With an angle of $60^{\circ}$ with QR, construct a ray QY and extend it to form a line YQY'.

Step 3: Now, cut off a line segment QS equal to 2 cm from QY' and join RS.

Y

Y'

Step 4: Draw a perpendicular bisector of RS which intersects QY at a point P.

Step 5: Join PR to get the required triangle $\triangle PQR.$

The steps of construction to be followed:

Step 1: For given $XY+YZ+ZX = 11 cm$ , a line segment $PQ =11 cm$ is drawn.

Step 2: At points, P and Q angles of $\angle RPQ = 30^{\circ}$ and $\angle SQP =90^{\circ}$ are constructed respectively.

Step 3: Now, bisects the angle RPQ and SQP. The bisectors of these angles intersect each other at a point X.

Step 4: Construct the perpendicular bisector of PX and QX, name them as TU and WV respectively.

Step 5: Let the bisector TU intersect PQ at Y and bisector WV intersect PQ at Z. Then XY and ZY are joined.

Therefore, $\triangle XYZ$ is the required triangle.

The steps of construction to follow:

Step 1: Draw a ray BX and Cut off a line segment $BC=12cm$ from the ray.

Step 2: Now, construct an angle $\angle XBY = 90^{\circ}$ .

Step 3: Cut off a line segment BD of length 18 cm on BY. Then join the CD.

Step 4: Now, construct a perpendicular bisector of CD which intersects BD at A and AC is joined.

Thus, the constructed triangle is ABC.

## Summary Of Class 9 Constructions NCERT Solutions

• Constructions in geometry are the methods of drawing geometric shapes and figures using a straightedge and a compass.

• Some basic constructions include drawing a line parallel to a given line through a given point, bisecting a given line segment, constructing the perpendicular bisector of a given line segment, and drawing an angle of a given measure.

• To construct a perpendicular bisector of a given line segment AB, we draw a circle with A and B as its endpoints, then draw another circle with the same radius and center at B. The intersection of the two circles gives the midpoint M of AB. The line passing through M and perpendicular to AB is the perpendicular bisector of AB.

• To construct an angle of a given measure, we first draw a ray with an endpoint O. We then place the compass at O and draw an arc intersecting the ray at point A. We then draw another arc with the same radius and center at A. We mark the point of intersection of these two arcs as B. The angle ∠AOB is the required angle.

• Constructions are useful in various fields, such as architecture, engineering, and art.

• The accuracy of constructions depends on the precision of the tools used and the skill of the person performing the construction.

• The constructions discussed in this chapter are important in higher-level geometry, and they form the basis for several advanced constructions and theorems.

Here students can find class 9 maths ch 11 question answer using exercise link given below.

## NCERT solutions for class 9 maths chapter wise

 Chapter No. Chapter Name Chapter 1 Number Systems Chapter 2 Polynomials Chapter 3 Coordinate Geometry Chapter 4 Linear Equations In Two Variables Chapter 5 Introduction to Euclid's Geometry Chapter 6 Lines And Angles Chapter 7 Triangles Chapter 8 Quadrilaterals Chapter 9 Areas of Parallelograms and Triangles Chapter 10 Circles Chapter 11 Constructions Chapter 12 Heron’s Formula Chapter 13 Surface Area and Volumes Chapter 14 Statistics Chapter 15 Probability

## NCERT solutions for class 9 subject wise

 NCERT Solutions for Class 9 Maths NCERT Solutions for Class 9 Science

How to use NCERT solutions for class 9 maths chapter 11 Constructions

• Understand the basic construction of angles using the previous class book.
• Learn the constructional process of triangles and rectangles.
• Go through some solved examples to understand the solutions pattern.
• Apply the concepts learned in the practice exercises.
• If you feel trouble in solving any problem then take the assistance of NCERT solutions for class 9 maths chapter 11 Constructions.

## NCERT Books and NCERT Syllabus

1. What are the important topics in construction maths chapter 11 class 9 ?

Basic Construction and Construction of triangles are two important topics in this chapter. Students can priorities the important chapters according to the NCERT syllabus and practice them to get good score well in the exam. practicing These NCERT Solutions will help students to command the concepts and provide confidence during the exam which ultimately lead to score well in the exams.

2. Where can I find the complete solutions of chapter 11 maths class 9 ?

Here students can get NCERT solutions for class 9 . these solutions are prepared by expert team at Careers360 conidering the students demand and thus are very beneficial for students. After practicing these solutions and problems you will get in-depth understanding of the concepts.

3. What is the significance of NCERT Solutions for Class 9 Maths Chapter 11 in terms of board exam preparation?

NCERT Solutions for ch 11 maths class 9 are a valuable resource that offers comprehensive information and understanding of every concept, which enables students to solve all types of questions, regardless of their level of difficulty. Consistent practice is crucial to learning and scoring well in Mathematics. Therefore, a wide range of questions, along with their solutions, shortcut techniques, and detailed explanations, are provided to practice any concept thoroughly.

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Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.

Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available

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
##### 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
##### 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

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
##### 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

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
##### 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