Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

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 is an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both sides.
(iv) If two circles are equal, then their radii are equal.
(v) In figure, if AB = PQ and PQ = XY, then AB = XY

Solution:
(i) False. In a single point, the infinite number of lines can pass through it.

(ii) False. For two distinct points only one straight line is passing.

(iii) True.

(iv) True.      [∵ Radii of congruent (equal) circles are always equal]
(v) AB = PQ  …(i)
PQ = XY   …(ii)
⇒ XY = PQ
From Eq. (i) and (ii), we get AB = XY

 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) a radius of a circle
(v) square
Solution:
(i) Parallel lines: Two lines in a plane are said to be parallel if they have no point in common.

In the figure, x and y are said to be parallel because they have no point in common and we write, x//y.
Here, the term point is undefined.

(ii) Perpendicular lines: Two lines in a plane are said to be perpendicular if they intersect each other at one right angle.

In the figure, P and Q are said to be perpendicular lines because they intersect each other at 90° and we write Q⊥P.
Here, the term one right angle is undefined.

(iii) Line segment: The definite length between two points is called the line segment.

In the figure, the definite length between A and B is line segment and represented by .
Here, the term definite length is undefined.

(iv) Radius of a circle: The distance from the center to a point on the circle is called the radius of the circle.
In the adjoining figure, OA is the radius.

Here, the term, point, and center are undefined.

(v) Square: A square is a rectangle having same length and breadth. Here, the terms length, breadth, and rectangle are undefined.

 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
(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 several undefined terms which the student should list. They are consistent because they deal with two different situations-
(i) says that the given two points A and B, there is a point C lying on the line in between them;
(ii) says that given A and B, we can take C not lying on the line through A and
These ‘postulates’ do not follow from Euclid’s postulates. However, they follow from axiom stated as 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 proves that AC =  AB. Explain by drawing the figure.
Solution:
Given, a point C lies between two points A and B such that AC – BC.

On adding AC to both sides, we get AC + AC = BC + AC
⇒ 2AC = AB ⇒ AC = AB Hence Proved

 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:
Here, C is the mid-point of line segment AB, such that

Let there are two mid-points C and C′ of AB.
AC = AB
AC′ = AB
⇒ AC’=AC
Which is only possible, when C anc C′ coincide.
⇒ Points C and C′ are identical.
Hence, every line segment has one and only one mid-point.

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

Solution:
According to axiom 5, we have the whole is greater than the part, which is a universal truth.
Let a line segment PQ = 8 cm. Consider a point R in its interior, such that PR = 5 cm.

Clearly, PR is a part of the line segment PQ and R lies in its interior. So, PR is smaller than PQ.
Hence, the whole is greater than its part and this is true for anything in any part of the world.
Note: This question is not about the fifth postulate.