NCERT Solutions for Class 10 Maths Chapter 11 Constructions

You can opt for Chapter 11 - Constructions NCERT Solutions for Class 10 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 10 Maths

• Chapter 1 - Real Numbers

• Chapter 2 - Polynomials

• Chapter 3 - Pair of Linear Equations in Two Variables

• Chapter 4 - Quadratic Equations

• Chapter 5 - Arithmetic Progressions

• Chapter 6 - Triangles

• Chapter 7 - Coordinate Geometry

• Chapter 8 - Introduction to Trigonometry

• Chapter 9 - Some Applications of Trigonometry

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Areas Related to Circles

• Chapter 13 - Surface Areas and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Constructions:   

Chapter 11 Constructions Class 10 contains four exercises, where the chapter starts with an introductory part  followed by section 11.2. In the first exercise, you would learn how to make the divisions in the line segment using geometrical instruments, and further in the second exercise, you will understand what is tangent and how to construct it on the circle followed by an exercise consisting of some practice questions for your in-depth understanding of the chapter.

Exercise 11.1 - Division of a line segment.

Exercise 11.2 - Construction of a tangent to a circle.

Concept of Division of the Line Segment:

To divide a line segment into the ratio of 3:2 internally.

Steps of Construction:

1. Draw a line segment AB of given length by using a ruler.

2. Draw a ray AX making an acute angle with AB.

3. Along AX mark off (m+n) points A1, A2, A3,....., Am+n such that AA1 = A1A2 = A2A3 = ……= Am+1Am+n.

4. Join BAm+n.

5. Through the point Am, draw a line parallel to Am+ B by making an acute angle equal to ∠ AAm+n B at Am.

6. Suppose there is a point P, and this line meets AB at P.

7. The line so obtained is the required point that divides the line segment AB into the ratio of m:n.

The figure so obtained is shown below:

Concept of Similar Triangles:

The criteria for the triangles to be similar to each other?

We have learned about the similarity of two triangles by using basic proportionality theorem, which states:

Theorem 1: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the triangles are similar. This can be referred to as angle-angle-angle or A-A-A criterion of the two similar triangles. 

Theorem 2: If in two triangles, sides of one triangle are proportional (that is in the same ratio) of the sides of the other triangle, then the corresponding angles are equal and hence the two triangles are similar. This can be referred to as side-side-side or S-S-S criterion of the two similar triangles.

Theorem 3: If an angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar to each other. This is referred to as side-angle-side or S-A-S criterion of the two similar triangles.

To understand how such similar triangles are constructed, let’s see the step-by-step explanation of the concept by our experts.

Example: Construct a triangle similar to a given triangle with sides 6 cm, 7 cm, and 8 cm and whose sides are 5/ 7th of the corresponding sides of the given triangle.

Solution.

Three sides of the given triangle are 6 cm, 7 cm and 8 cm.

Steps of Construction:

1. Draw a line segment BC = 7 cm.

2. With B as a center and with radius 6 cm, draw an arc.

3. With C as a center and with radius  cm, draw another arc, intersecting the previously drawn arc at A.

4. Join AB and AC. Then, △ABC.

5. Below BC, make an acute angle ∠ CBX.

6. Along BX, mark off seven points X1, X2, X3,.....,X7 such that XX1 = X1X2 = X2X3=…..= X6X7.

7. Join X7 to C and draw a line via X5 parallel to X7C, intersecting the extended line segment BC at C’.

8. Draw a line via C’ parallel to CA intersecting the line segment BA at A’. Then △ A’B’C’ is the required triangle which is similar to the given triangle △ABC.

The figure after following the above steps is drawn below:

Concept of the construction of a tangent through a point lying outside to the circle.

Our experts have explained this concept by solving a question step-by-step to make you understand the concept easily. 

To Construct a tangent to a circle from an external point.

Steps for construction

1. Take any given circle of some radius ‘r’.

2. Take a given circle and a point P outside the circle. O is the center of the circle.

3. Join OP.

4. Bisect OP and get its midpoint M.

5. Draw a circle with center M and radius = PM = MO.

6. Circle drawn meets the given circle at Q above PO and at Q’ below PQ’.

7. PQ and PQ’ are the required tangents drawn to the circle from the point P. We observe that PQ = PQ’.

Here is the constructed tangent to the circle:

Persistence leads to perfection - NCERT Solutions

“Success is the consequence of flawlessness, diligence, learning from failure, loyalty, and perseverance”.

• All the people are not Mathematicians by birth, still, they have the ability to produce divergent solutions to any problem. Some take time to do so, and they come out with flying colors, what makes them different from others?

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• It will give you step-by-step solutions to questions in-depth to make you apprehend the concepts finer and clear your confusion.

• In this way, students will have a grasp over the basic concepts finer and faster.

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• Class 10 chapter 11 solutions are prepared in a concise manner by experts with illustrative examples to help you all in scoring good marks in your board exams. 

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Class 10 Chapter 11 Marks Distribution Format for Exam 2020 as per the Latest CBSE Guidelines:

This chapter will have four sections, namely: A, B, C, D

In Class 10 Maths, Paper will be of 40 Questions Divided into Four Sections:

Total questions: 01 (One long answer-type question)

Total marks: 04 marks for the Chapter Constructions as per the latest CBSE syllabus.

About the Chapter

Construction appears in formal geometry as integral parts of geometry, as important as theorems. A geometric construction helps you to draw shapes when you don’t have a computer to do it for you. Some Maths problems are better solved when you draw the actual object using the constructing instruments without any error in your measurements, and obtain the figure you expect especially in solving any Physics or a Maths problem which requires a diagram for the same. Chapter 11 Constructions of Class 10 Maths NCERT syllabus is divided into two sections and two exercises:

Section 11.1

The chapter starts with a short introduction to the topics you learned in the previous classes and relates the previously learned concepts into this chapter.

Section 11.2

Exercise 11.1: This chapter further proceeds with the Construction of two topics mentioned below:

• Division on a line segment in a given ratio a:b.

• Draw a triangle similar to the given triangle with a given scale factor in the ratio of m: n.

Exercise 11.1: This exercise contains 7 questions on divisions of a line segment and drawing a triangle similar to the given triangle.The first construction taught here is the Division of the line segment and activities to perform these constructions in your real-life too. Such examples are based on real-life objects to make you feel connected with the real world.

In this section, you will understand the step-by-step explanation of the construction of the division of any given line segment. Our faculties have given the justification of the construction at each step via illustrative examples to give you a crystal clear understanding of the concepts. On proceeding further, you will learn to apply the property of similarity in two triangles in construction.  

You will study how to draw a triangle similar to the given triangle to get an insight into the logic of how similar triangles are drawn.

You will be applying the Basic proportionality theorem (Thale's theorem) to understand the concept of similarity between two triangles.

The construction of such a similar triangle will be done on the basis of the concept of the given scale factor.

Section 11.3

Exercise 11.2: This exercise contains 7 questions related to construction and drawing the given object by degrees following the points given in the question. 

In these questions, a crystal clear justification of each step is given by the teacher to make the understanding of the concept easier and would take no time for a student to retain the concepts even in the future.

This exercise contains questions that are related to the construction of tangents to a circle.

Our faculties have explained the concept of tangents and applied the knowledge of the same to make you understand how to construct the tangents to a circle from a point outside it. 

Furthermore, this exercise consists of different questions related to the constructions such as drawing of tangents to the circle, measuring its length, drawing a line segment, constructing a circle with a fixed radius ‘k’. Then constructing the tangent to that drawn circle.

Through this exercise, you would be able to implement the application of tangents you have learned in the previous classes into the construction of a tangent to any given circle.

Summary

In this chapter you learned about:

• How to make the use of previous concepts into chapter 11?

• How to use this concept in chapter Heights and distances? 

• To make divisions on the line segment into given ratio of m:n

• To draw a triangle similar to the given triangle in the ratio with any of two cases:

1. When scalar factor < 1

2. When scalar factor > 1

• To draw a tangent through a point lying outside the circle.

What do you Need to know Before you Start Studying this Chapter?

In previous classes, you would have learned the basics of the Constructions and the properties of the similarity between two triangles. This includes the following: 

• Constructing a perpendicular bisector from the line segment 

• Drawing a line perpendicular from a point to the line segment.

• Constructing an angle bisector.

• Determining the midpoint of the line segment.   

• The similarity in two triangles on the basis of their attributes

• The reason for having this exercise is to get a deep understanding of the chapter so that you get the practice of this chapter thoroughly which will be useful in the construction of a diagram for the topic heights and distances.

You need to have a crystal clear understanding of all these concepts in order to start with Class 10 Chapter 11 Constructions. Since you are done with these concepts, in this chapter you will apply all the logics and go-ahead to an upgraded level of Chapter Constructions. You will be understanding how to divide the given line both internally and externally in the ratio of m: n.

You will also understand how to construct a triangle similar to the given triangle by using the concept of similarity you have learned in the previous classes. You will also learn how to construct a tangent to the circle to a given point lying outside the circle.

Best of Both Worlds

Ever since the world came into existence, people have been toiling hard to get the ‘Best thing since sliced bread’ and working vigorously to Construct one or the other thing every day and this is continuously going on and will continue.

Whatever you construct by your innovation, directly has an application of  ‘Construction.’ It would always help you in real-life situations too.

Humans are sophisticated scientists and would continue to investigate something new each day.

So, why shouldn’t a little scientist like ‘you’ do something innovative 

each day?  

History of Construction:

The history of construction has a place in  myriads of fields like structural engineering, civil engineering and reckons on other arms of science like archaeology, history, and architecture, to inquire how the builders lived and canned their acquisitions. Those fields let us dissect constructed buildings and other structures built since prehistory, the tools used, and the different uses of building materials.

History of a building is germinating by different vogue in time, marked by few key maxims: enduringness of the materials used, the increasing of height and span, the degree of control exercised over the interior environment, and finally the energy accessible to the construction process.

Copper and Bronze Age Construction

• The Copper Age is the archaeozoic constituent of the Bronze Age. Copper came into utilization ahead 5,000 BC with the addition of bronze around 3,100 BC, Copper and bronze were used for the assonant types of tools as stone such as axes and chisels. 

• Bronze was molded into desired shapes and if discredited could be recast. A novel tool formulated in the copper age is the saw. People employed copper and bronze points for operating soft stone extracting quarrying blocks and fashioning rock-cut architecture.

• All the way through the Bronze Age the corbelled arch came into use such as for beehive tombs. The wheel came into use. Subsequently, heavy loads were moved on boats, sleds (a primitive sled), or on rollers. 

• The Egyptians started out building stone temples accompanying the post and lintel construction orderliness and the Greeks and Romans adopted this style.

Iron Age Construction

• The Iron Age is a cultural period from approximately 1200 BC to 50 BC accompanying the far-flung use of iron for tools and weapons. Iron is not much harder than bronze but by appending carbon iron turns to steel which was being produced after about 300 BC. Steel can be endured and toughened bringing forth a carnassial, imperishable cutting edge. A novel woodworking tool allotted by the use of steel is the hand-plane.