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

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

Edited By Safeer PP | Updated on Jul 18, 2022 02:10 PM IST

The subject of Euclid's Geometry and its axiom is covered in NCERT Solutions for Class 9 Maths exercise 5.1. Euclid's Geometry is the study of geometry based on undefined words such as points, lines, and planes of flat spaces, which was developed by Euclid, the Father of Geometry. Euclid's geometry notably spoke about the shape, size, and position of solid shapes. In NCERT book exercise 5.1 Class 9 Maths as well as different concepts related to the shapes such as the surface, straight or curved lines, points, and so on... In terms of geometry, there are seven axioms that were given by Euclid.

This Story also Contains
  1. Introduction To Euclid's Geometry Class 9 Chapter 5 Exercise: 5.1
  2. More About NCERT Solutions for Class 9 Maths Exercise 5.1
  3. NCERT Solutions of Class 10 Subject Wise
  4. Subject Wise NCERT Exemplar Solutions

• Axiom -1: Things that are equal to each other are also equal to each other.

• Axiom -2: When equals are added together, the result is the same.

• Axiom -3: If you subtract equals from equals, the result is equal.

• Axiom -4: Everything in the same place at the same time is equal.

• Axiom -5: The total of the parts exceeds the total of the parts.

• Axiom -6: Doubles of the same thing are doubles of the same thing.

• Axiom -7: Things that are halves of the same thing are the same thing.

NCERT solutions for Class 9 Maths chapter 5 exercise 5.1 consists of seven questions based on the fundamental terminology, theorems, facts, and definitions of points, lines, circles, radius, angles, planes, and other concepts. In this Class 9 Maths chapter 5 exercise 5.1, the ideas linked to Euclid's geometry are thoroughly discussed. The following exercise is included along with Class 9 Maths chapter 5 exercise 5.1.

Introduction To Euclid's Geometry Class 9 Chapter 5 Exercise: 5.1

Q1 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.

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

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

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

(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

1640082731275

Answer:

i) FALSE
Because there is the infinite number of lines that can be passed through a single point. As shown in the diagram below
1640082806946 ii) FALSE
Because only one line can pass through two distinct points. As shown in the diagram below
1640082824490 iii) TRUE
Because a terminated line can be produced indefinitely on both sides. As shown in the diagram below
1640082837386 iv) TRUE
Because if two circles are equal, then their centre and circumference will coincide and hence, the radii will also be equal.

v) TRUE
By Euclid’s first axiom things which are equal to the same thing, are equal to one another

Q2 (i) 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?
(i) parallel lines

Answer:

Yes, there are other terms that are needed to be defined first which are:

Plane: A plane is a flat surface on which geometric figures are drawn.

Point: A point is a dimensionless dot that is drawn on a plane surface.

Line: A line is the collection of n number of points that can extend in both directions and has only one dimension.
i) Parallel line:-
If the perpendicular distance between two lines is always constant and they never intersect with each other in a plane. Then, two lines are called parallel lines.

Q2 (ii) 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? (ii) perpendicular lines

Answer:

Yes, there are other terms that are needed to be defined first which are:

Plane: A plane is a flat surface on which geometric figures are drawn.

Point: A point is a dimensionless dot which is drawn on a plane surface.

Line: A line is the collection of n number of points which can extend in both the directions and has only one dimension.
ii) perpendicular line:-
If two lines intersect with each other and make a right angle at the point of intersection. Then, two lines are called perpendicular lines.

Q2 (iii) 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? iii) line segment

Answer:

Yes, there are other terms that are needed to be defined first which are:

Plane: A plane is a flat surface on which geometric figures are drawn.

Point: A point is a dimensionless dot that is drawn on a plane surface.

Line: A line is a collection of n number of points that can extend in both directions and has only one dimension.
iii) line segment: -
A straight line with two end points that cannot be extended further and has a definite length is called line segment

Q2 (iv) 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? iv) radius of circle

Answer:

iv) Radius of the circle: -
The distance between the centre of the circle and any point on the circumference of the circle is called the radius of a circle.

Q2 (v) 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? v) square

Answer:

v) Square:-
A square is a quadrilateral in which all the four sides are equal and each internal angle is a right angle.

To define the square, we must know about quadrilateral.

Q3 Consider two ‘postulates’ given below:

(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.

(ii) There exist at least three points that are not on the same line.

Do these postulates contain any undefined terms? Are these postulates consistent?

Do they follow from Euclid’s postulates? Explain.

Answer:

There are various undefined terms in the given postulates.:

1) There is no information given about the plane whether the points are in the same plane or not.

2) There is an infinite number of points lying in a plane. But here the position of point C has not specified whether it lies on the line segment joining AB or not.

Yes, these postulates are consistent when we deal with these two situations:

(i) Point C is lying in between and on the line segment joining A and B.

(ii) Point C does not lie on the line segment joining A and B.

No, they don’t follow from Euclid’s postulates. They follow the axioms.

Q4 If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2 AB. Explain by drawing the figure.

Answer:

It is given that
AC = BC
Now,
1640082867954 In the figure given above, AB coincides with AC + BC.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB
Now,
2AC = AB (\because AC = BC )
Therefore,
AC = \frac{1}{2}AB
Hence proved.

Q5 In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Answer:

1640082899454 Let's assume that there are two midpoints C and D
Now,
If C is the midpoint then, AC = BC
And
In the figure given above, AB coincides with AC + BC.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB
From this, we can say that
2AC = AB -(i)

Similarly,
If D is the midpoint then, AD = BD
And
In the figure given above, AB coincides with AD + BD.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AD + BD = AB
From this, we can say that
2AD = AB -(ii)
Now,
From equation (i) and (ii) we will get
AD = AC
and this is only possible when C and D are the same points
Hence, our assumption is wrong and there is only one midpoint of line segment AB.

Q6 In Fig. 5.10, if AC = BD, then prove that AB = CD.

1640082935536

Answer:

From the figure given in the problem,
We can say that
AC = AB + BC and BD = BC + CD
Now,
It is given that AC = BD
Therefore,
AB + BC = BC + CD
Now, According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal. Subtracting BC from both sides.
We will get
AB + BC - BC = BC + CD - BC
AB = CD
Hence proved

Q7 Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

Answer:

Axiom 5 states that the whole is greater than the part.
Lets take A = x + y + z
where A , x , y , z all are positive numbers
Now, we can clearly see that A > x , A > y , A > z
Hence, by this we can say that the whole (A) is greater than the parts. (x , y , z)

More About NCERT Solutions for Class 9 Maths Exercise 5.1

Euclid's postulate was well discussed in Class 9 Maths chapter 5 exercise 5.1. There are five of Euclid's postulates in terms of geometry. They are,

• Postulate -1: For any two points, a straight line segment may be drawn.

• Postulate -2: To make a line, a line segment can be extended in any direction.

• Postulate -3: To describe a circle with any centre and radius, a circle with any centre and radius can be drawn.

• Postulate -4: Every right angle is equal to every other right angle.

• Postulate -5: If two lines are intersected by a third in such a way that the sum of the inner angles on one side is less than two right angles, the two lines will intersect on that side indefinitely.

Also Read| Introduction To Euclid's Geometry Class 9 Notes

Benefits of NCERT Solutions for Class 9 Maths Exercise 5.1

• NCERT solutions for Class 9 Maths exercise 5.1, practising these questions will help us to get well-versed with the key fundamentals of Euclid’s geometry.

• If we go through the NCERT solution for Class 9 Maths chapter 5 exercise 5.1, we can grasp the core geometry concepts by learning the basic properties of points, lines, circles, radius, angles, planes, etc… and also we can attain the basic knowledge required for learning many advanced topics.

• On solving the questions of exercise 5.1 Class 9 Maths, will enable us to focus and develop the skills required to score well in exams and also along solving the questions of Class 9 Maths exercise 5.1 should memorize the definitions, axioms and postulates.

Also see-

NCERT Solutions of Class 10 Subject Wise

Subject Wise NCERT Exemplar Solutions

Frequently Asked Question (FAQs)

1. Define Euclid’s geometry , as per NCERT solutions for Class 9 Maths chapter 5 exercise 5.1 ?

Euclid's geometry is defined as the study of geometry based on undefined notions such as points, lines, and planes of flat spaces, according to NCERT solutions for Class 9 Maths chapter 5 exercise 5.1

2. Who introduced Euclid’s geometry?

Euclid’s geometry was introduced by Euclid. 

3. Euclid is the Father of ______.

Euclid is the Father of Geometry.

4. Define perpendicular lines , According to NCERT solutions for Class 9 Maths chapter 5 exercise 5.1.

The lines that intersect each other in a plane at right angles are said to be perpendicular to each other, according to NCERT solutions for Class 9 Maths chapter 5 exercise 5.1

5. The number of axiom of Euclid’s geometry is ______

The number of axioms of Euclid’s geometry is 7.

6. Rani is of the same age as vani . Sudha is also of the same age as vani . State the correct Euclid’s axiom that illustrates the relative ages of Rani and Sudha

The correct Euclid’s axiom that illustrates the relative ages of Rani and Sudha is the first axiom. According to the first axiom, Things that are equal to the same thing are equal to one another.

7. How many lines can pass through a given point?

There are an endless number of lines that can pass through a given spot.

8. Things that are similar to one another are referred to as _____

Things that are similar to one another are referred to as equal. 

Reason: According to axiom 4, things that are at the same place at the same time are equal.

9. A straight line segment can be drawn for any two given points is stated in the form of _______

A straight line segment can be drawn for any two given points as stated in the form of a postulate

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