NCERT Solutions for Class 9 Maths Chapter 5: Introduction To Euclid's Geometry - Exercise 5.1

Exercise 5.1

1. Which of the following statements are true and which are false?  give reasons for your answers.

(i) Only one line can pass through a single point.

Ans: False 

Because an endless number of lines can pass through a single point \[\text{ }\!\!'\!\!\text{  P  }\!\!'\!\!\text{ }\] below. There exists an infinite number of lines travelling through a single point \[\text{P}\], as shown in the diagram below.

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(ii) There are an infinite number of lines which pass through two distinct points

Ans: False 

Only one line can pass through two points. There is only one single line that can travel between two separate points \[\text{P}\] and \[\text{Q}\], as shown in the following diagram.

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(iii) A terminated line can be produced indefinitely on both the sides

Ans: True  

• On both the sides, a terminated line can be produced indefinitely.

• Assume that \[\text{AB}\] is a terminated line. It can be seen that it can be produced indefinitely on both the sides.

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(iv) If two circles are equal, then their radii are equal.

Ans: True 

If two circles are equal, then the centers and circumferences of the two circles are the same, the radii will be equal.

(v) In the following figure, if $\text{AM=PQ}$ and $\text{PQ=XY}$ ,then $\text{AB=XY}$

Ans: True 

• It is assumed that \[\text{AB}\] and \[\text{XY}\] are two terminated lines (Line segments) and that they are both equal to \[\text{PQ}\], a third line.

• Euclid's first axiom stats that things which are equal to the same thing are equal to one another.

• Therefore, the lines \[\text{AB=PQ}\] and \[\text{PQ=XY}\] , hence $AB=XY$ will be equal to each other.

2. Give a definition for each of the following terms. Are there other terms that need to be defined first ? what are they, and how might you define them ?

1. Parallel lines

2. Perpendicular lines

3. Line segment

4. Radius of  a circle

5. Square

Ans:

The following terms are required for the desired definition:

• Point:

• A point can be approximated by a little dot formed with a sharp pencil on a sheet of paper.

• A point does not have any dimensions; it simply has a position.

• Line: 

• A straight line made by folding a piece of paper, a straight string pulled at both ends, and the edge of a ruler are all examples of geometrical lines.

• The basic concept about a line is that it should be straight and that it should extend in definitely in both the direction 

• Plane:

• Close examples of planes include the smooth surface of a wall or the smooth surface of a piece of paper.

• Ray:

• A ray\[\text{AB}\] is a segment of line l that has only one end point \[\text{A}\] and contains the point \[\text{B}\].

• A point can be approximated by a little dot formed with a sharp pencil on a sheet of paper.

• A point does not have any dimensions; it simply has a position.

• A straight line made by folding a piece of paper, a straight string pulled at both ends, and the edge of a ruler are all examples of geometrical lines.

• The fundamental principle of a line is that it should be straight and continue indefinitely in both directions.

• Close examples of planes include the smooth surface of a wall or the smooth surface of a piece of paper.

• A ray is a segment of line l that has only one end point and contains the point.

• The union of two non-collinear rays with a common beginning point is called an angle .

• Circle:

• In a plane, a circle is the collection of all points whose distance from a fixed point is constant. 

• The fixed point is called the centre of the circle

• Quadrilateral:

• A closed figure made of four lines segment is called quadrilateral.

(i) Parallel lines

Ans: 

• Parallel lines are those in which the perpendicular distance between two lines is always the same.

• To put it another way, parallel lines are lines that never cross one other.

• To define parallel lines, we need to know about the point, the lines, and the distance between the lines and the intersection point.

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(ii) Perpendicular lines

Ans: 

• If two lines intersect each other at \[\text{90 }\!\!{}^\circ\!\!\text{ }\] then these are called perpendicular lines.

• Before defining perpendicular lines, we must first define the line and the angle.

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(iii) Line segment 

Ans: 

• A line segment is a straight line drawn from one point to another point.

• To define a line segment, we must first understand what a point and a line segment are.

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(iv) Radius of a circle

Ans: 

• It is distance between the center of a circle to any point lying on the circle.

• We must understand point and circle in order to define the radius of a circle.

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(v) Square 

Ans: 

• A square is a quadrilateral with all sides equal in length and all angles measuring \[\text{90 }\!\!{}^\circ\!\!\text{ }\].

• To define square, we must know about quadrilateral, side, and angle.

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3. Consider the two 'postulates' given below:

1. Given any two distinct points A and B, there exists a third point C, which is between A and B 

2. There exists at least three points that are not on the same line.Do these postulates contain any undefined terms ? Are these postulates consists ? Do they follow from Euclid's postulates\[\text{?}\]Explain.

Ans: 

• In the given postulates, there are several undefined terms.

• Because the above postulates pertain to two different situations, they are consistent.

• Furthermore, any assertion that contradicts a well-known axiom or postulate is impossible to infer.

• The postulates of Euclid do not lead to these conclusions.

• They follow from the axiom, "Given two distinct points, there is a unique line that passes through them"

4. If a point C lies between two points A and Bsuch that AC=BC then prove that \[\text{AC=}\frac{\text{1}}{\text{2}}\text{AB}\] Explain by drawing the figure.

Ans:

From the Figure,

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Given that \[\text{AC=BC}\]

And point $C$ lies between two points \[\text{A}\] and \[\text{B}\]

Consider  c 

Adding \[\text{AC}\] on both sides we get

\[\text{AC+AC=BC+AC}\]

\[\Rightarrow \text{2AC=BC+AC}\]

Here we have \[\text{BC+AC=AB}\]

\[\Rightarrow \text{2AC=AB}\]

\[\Rightarrow \text{AC=}\frac{\text{1}}{\text{2}}\text{AB}\]

5.  In above question, point C is called a midpoint of line segment AB, prove that every line segment has one and only mid point

Ans:

From the Figure,

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Easier solution

Now to we will prove line \[\text{AB}\] has only one midpoint such that 

Consider we have two midpoint \[\text{C}\] and \[\text{D}\] of line segment \[\text{AB}\]

Thus 

\[\text{AD=DB}\]         ......\[\left( \text{1} \right)\]

\[\text{AC = CB}\]          ...... \[\left( \text{2} \right)\]

Now subtracting equation \[\left( \text{1} \right)\text{-}\left( \text{2} \right)\] we get

\[\text{AD - AC = DB - CB}\]

Using figure we have 

\[\Rightarrow \text{CD = -DC}\]

\[\Rightarrow \text{2CD = 0}\]

\[\Rightarrow \text{CD = 0}\]

Therefore \[\text{C}\] and \[\text{D}\]  coincides.

Hence required is proved.

Lengthy solution 

\[\text{AC = CB}\]          ...... \[\left( \text{1} \right)\]

Now adding \[\text{AC}\] on both sides of equation \[\left( \text{1} \right)\]  we get

\[\text{AC+AC = AC+CB}\]        ...... \[\left( \text{2} \right)\]

From the figure we have 

\[\text{AC+CB=AB}\]

Now from equation \[\left( \text{2} \right)\] we have

\[\text{2AC = AB}\]        ...... \[\left( \text{3} \right)\]

Similarly we have 

\[\text{2AD = AB}\]        ...... \[\left( \text{4} \right)\]

Now equalizing equation \[\left( \text{3} \right)\] and \[\left( \text{4} \right)\] we get

\[\text{2AC = 2AD }\]

\[\Rightarrow \text{AC = AD }\]

Therefore \[\text{C}\] and \[\text{D}\]  coincides.

Hence required is proved.

6. In the following figure, if AC = BD, then prove that AB = CD

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Ans:

We are asked to prove \[\text{AB = CD}\]

Let 

 \[\text{AB = CD}\]       ...... \[\left( \text{1} \right)\]

Now adding \[\text{BC}\] on both sides of equation \[\left( \text{1} \right)\]  we have 

\[\text{AB+BC = CD+BC}\]

From the figure we have 

\[\text{AC = BD}\]       ...... \[\left( \text{2} \right)\]

Now from the figure we have 

x\[\text{AC = AD - CD}\]        ...... \[\left( \text{3} \right)\]

\[\text{BD = AD - AB}\]        ...... \[\left( \text{4} \right)\]

Using above equation\[\left( \text{2} \right)\], \[\left( \text{3} \right)\] and  \[\left( \text{4} \right)\]  we have

\[\text{AD - CD = AD - AB}\]

\[\Rightarrow \text{- CD = - AB}\]

\[\Rightarrow \text{AB = CD}\]

Hence required is proved 

7. Why is Axiom in the list of Euclid's axioms, considered a 'universal truth'\[\text{?}\](Note that the question is not about the fifth postulate.)

Ans:

Axiom $\text{5}$  states that the whole is greater than the part.

This axiom is known as a universal truth because it holds true because it holds true in any field, and not just in the field of mathematics 

Let us take two cases one in the field of mathematics, and one other than that 

• Case one

• Let $\text{t}$ represent a whole quantity and only $\text{a,b,c}$ are parts of it.

• $\text{t = a + b + c}$

• Clearly $\text{t}$ will be greater than all its parts $\text{a,b,c}$.

• As a result, it is correct to say that the whole is greater than the part

• Case two

• Let us consider the continent Asia.

• Now consider the country of India, which is located in Asia.

• Although India is a part of Asia, it is also true that Asia is larger than India.

• As a result, we might conclude that the whole is greater than the part.

• This holds true in any corner of the world, making it a universal truth.

Class 9 Maths Chapter 5 – Exercise 5.1

Exercise 5.1 Class 9 Covers the Following Topics:

1. Introduction

2. Euclid’s Definitions, Axioms, and Postulates

3. Postulate 1

4. Postulate 2

5. Postulate 3

6. Postulate 4

7. Postulate 5

8. Exercise 5.1 questions and answers

Introduction

Geometry is one of the most important subjects in the field of Mathematics. It is used in many areas by many technical experts such as architects to create buildings that will last forever. This section gives the best introduction to Euclid’s geometry exercise 5.1 

The first section of NCERT Maths Class 9 chapter 5 exercise 5.1 focuses upon getting the basics right. It explains what is geometry and talks about its origination. It gives us insight into how the word was coined and how ancient civilizations all over the world used it to their various benefits. Through this chapter, the students will come to know the unsystematic methods of the early days of geometry and how it was nurtured by different cultures. It also informs the students of when and how the first mathematical proof was declared and how these chains of events had a strong influence on the famous Greek mathematician Euclid and his work.

Euclid’s Definitions, Axioms, and Postulates

The students learn in ex 5.1 Class 9 in detail about the works of the Greek mathematician Euclid. The chapter informs the students of the approach and methods used by Euclid to arrive at his Definitions. The chapter further explores the Definitions put forward by Euclid to make the students have a clear understanding of the concept. The chapter then moves on to explain the meaning of Axiom and Postulates. This Chapter gives students information on many concepts related to geometry in particular and Mathletics in general. This chapter is crucial for students’ acumen in the subject.

Exercise 5.1 Questions and Answers

Once you have completed studying the above-mentioned sections, you will have to complete 7 questions that are mentioned in Exercise 5.1 to access your understanding of the chapter. Make sure that you go through the questions carefully and then write the answers appropriately.

Postulate 1

This segment first informs the students of the very first Postulate which was stated by the great Greek mathematician Euclid. Its first states the exact Postulate then gets deeper in its details about things which have been said and which has not been said. It talks about how he came to this conclusion, the method he adopted and his process. The chapters also provide clear diagrams and drawings for the best understanding.

Postulate 2

This part talks about the second postulate stated by Euclid. It explores what the great mathematician meant by terminated lines. It also helps the student understand what happens when a line segment is extended on either side. Diagrams and drawings were given to help make the concept more transparent.

Postulate 3

This area of Class 9 Maths exercise 5.1 takes a further deeper journey in Euclid’s theories. It establishes many properties of circles. It elaborates if a circle can be drawn from any given centre or can it be also drawn from any given radius.

Postulate 4

This section of the chapter talks about a crucial principle of geometry in Mathematics. This chapter will teach you whether all right angles can be equal to one another. This theory is very crucial in Mathematics and has been widely used for further experiments thus helping shape the Mathematics we see today.

Postulate 5

In this segment theory about straight lines, is mentioned particularly the one Euclid strongly put forward. Through this segment, the student will learn whether it is possible for a straight line to fall on another straight line in any possible condition. And if so, happens in a given condition then where would the two straight lines meet. With this important explanation, this segment enables the student to master the subject.

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