NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry
Euclid Geometry Class 9- NCERT Solutions for Maths Chapter 5
Euclid Geometry Class 9 solutions are provided by SimplyAcad in a step-by-step way for all the exercises of the NCERT Solutions for Maths Chapter 5 as prescribed for Class 9 below. The exercises are designed to test the student’s understanding of the concept, the chapter holds a lot of significance as it is the basis of chapters that will be forming the syllabus of advanced mathematics. Therefore, keeping the fundamentals clear is equally important. Through the given solutions students will get an elaborate analysis of the chapter.
Euclid Geometry Class 9 Overview of the Exercises of NCERT Solutions Chapter 5
- Exercise 5.1: The first exercise includes true and false questions to check the basics of students, followed by drawing figures to explain and prove equations on a line, and proof related to the midpoint segment.
- Exercise 5.2 : The second exercise of the chapter includes a total of two questions of Euclid fifth postulate.
Euclid Geometry Class 9 : NCERT Solutions for Chapter 5 Exercise 5.1
Question 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.
(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 the following figure, if AB = PQ and PQ = XY, then AB = XY.
Solution:
(i) False. Since through a single point, infinite number of lines can pass. In the following figure, it can be seen that there are infinite numbers of lines passing through a single point P.
(ii) False. Since through two distinct points, only one line can pass. In the following figure, it can be seen that there is only one single line that can pass through two distinct points P and Q.
(iii) True. A terminated line can be produced indefinitely on both the sides.
Let AB be a terminated line. It can be seen that it can be produced indefinitely on both the sides.
(iv)True. If two circles are equal, then their centre and circumference will coincide and hence, the radii will also be equal.
(v) True. It is given that AB and XY are two terminated lines and both are equal to a third line PQ. Euclid’s first axiom states that things which are equal to the same thing are equal to one another. Therefore, the lines AB and XY will be equal to each other.
Question 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?
(i) Parallel lines
(ii) Perpendicular lines
(iii) Line segment
(iv) Radius of a circle
(v) Square
Solution:
(i) Parallel Lines
If the perpendicular distance between two lines is always constant, then these are called parallel lines. In other words, the lines which never intersect each other are called parallel lines.
To define parallel lines, we must know about point, lines, and distance between the lines and the point of intersection.
(ii) Perpendicular lines
If two lines intersect each other at
, then these are called perpendicular lines. We are required to define line and the angle before defining perpendicular lines.
(iii) Line segment
A straight line drawn from any point to any other point is called as line segment. To define a line segment, we must know about point and line segment.
(iv) Radius of a circle
It is the distance between the centres of a circle to any point lying on the circle. To define the radius of a circle, we must know about point and circle.
(v) Square
A square is a quadrilateral having all sides of equal length and all angles of same measure, i.e.,
To define square, we must know about quadrilateral, side, and angle.
Question 3: 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.
Solution:
There are various undefined terms in the given postulates.
The given postulates are consistent because they refer to two different situations. Also, it is impossible to deduce any statement that contradicts any well known axiom and postulate.
These postulates do not follow from Euclid’s postulates. They follow from the axiom, “Given two distinct points, there is a unique line that passes through them”.
Question 4:
If a point C lies between two points A and B such that AC = BC, then prove that AC = AB/2. Explain by drawing the figure.
According to the given statement, the figure will be as shown alongside in which the point C lies between two point A and B such that AC= BC.
Clearly, AC + BC = AB
AC + AC = AB [ AC= BC]
2AC = AB
AC = AB/2
Question 5: 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.
Solution:
According to the given statement, the figure will be as shown alongside in which the opint C lies between two point A and B such that AC= BC.
Clearly, AC + BC = AB
AC + AC = AB [ AC= BC]
2AC = AB
AC = AB/2
Question 6:
In the following figure, if AC = BD, then prove that AB = CD.
From the figure, it can be observed that
AC = AB + BC
BD = BC + CD
It is given that AC = BD
AB + BC = BC + CD (1)
According to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal.
Subtracting BC from equation (1), we obtain
AB + BC − BC = BC + CD − BC
AB = CD
Question 7: 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.)
Solution:
As statement is true in all the situations. Hence, it is considered a ‘universal truth.’
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