Important Questions for Class 9 Chapter 5-Introduction to Euclid's Geometry

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Important questions based on NCERT syllabus for Class 9 Chapter 5 - Introduction to Euclid’s Geometry:

Question 1: Consider the two ‘postulates’ given below:
(i) Given any two distinct points A and B, there exists a third point C, which is between A and B.
(ii) There exists 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 2: In the above question, point C is called a mid-point of line segment AB, prove that every line segment has one and only one mid-point.

Solution: Let there be two mid-points, C and D.
Line
C is the mid-point of AB.
AC = CB
AC + AC = BC + AC (Equals are added on both sides) ... (1)
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
∴ BC + AC = AB ... (2)
It is also known that things which are equal to the same thing are equal to one another. Therefore,
from equations (1) and (2), we obtain
AC + AC = AB
⇒ 2AC = AB ... (3)
Similarly, by taking D as the mid-point of AB, it can be proved that 2AD = AB ...(4)
From equation (3) and (4), we obtain
2AC = 2AD (Things which are equal to the same thing are equal to one another.)
⇒ AC = AD (Things which are double of the same things are equal to one another.) This is possible
only when point C and D are representing a single point.
Hence, our assumption is wrong and there can be only one mid-point of a given line segment.

Question 3: 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:
Axiom 5 states that the whole is greater than the part. This axiom is known as a universal truth 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 I:
Let t represent a whole quantity and only a, b, c are parts of it.
t = a + b + c
Clearly, t will be greater than all its parts a, b, and c.
Therefore, it is rightly said that the whole is greater than the part.
Case II:
Let us consider the continent Asia. Then, let us consider a country India which belongs to Asia. India is a part of Asia and it can also be observed that Asia is greater than India.
That is why we can say that the whole is greater than the part.
This is true for anything in any part of the world and is thus a universal truth.