Chapter 5 Introduction to Euclid Geometry
Ex 5.1
Ex 5.1 Class 9
Maths 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 figure, if AB – PQ and PQ = XY, then AB = XY.
Solution:
(i) False
Reason : If we mark a point O on the surface of a paper. Using pencil and
scale, we can draw infinite number of straight lines passing
through O.
(ii) False
Reason: In the following figure, there are many straight lines passing through
P. There are many lines, passing through Q. But there is one and only one line
which is passing through P as well as Q.
(iii) True
Reason: The postulate 2 says that “A terminated line can be produced
indefinitely.”
(iv) True
Reason: Superimposing the region of one circle on the other, we find them
coinciding. So, their centres and boundaries coincide.
Thus, their radii will coincide or equal.
(v) True
Reason: According to Euclid’s axiom, things which are equal to the same thing
are equal to one another.
Ex 5.1 Class 9
Maths 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:
Yes, we need to have an idea about the terms like point, line, ray, angle,
plane, circle and quadrilateral, etc. before defining the required terms.
Definitions of the required terms are given below:
(i) Parallel Lines:
Two lines l and m in a plane are said to be parallel, if they have no common
point and we write them as l ॥ m.
(ii) Perpendicular Lines:
Two lines p and q lying in the same plane are said to be perpendicular if they
form a right angle and we write them as p ⊥ q.
(iii) Line Segment:
A line segment is a part of line and having a definite length. It has two
end-points. In the figure, a line segment is shown having end points A and B.
It is written as
(iv) Radius of a circle:
The distance from the centre to a point on the circle is called the radius of
the circle. In the figure, P is centre and Q is a point on the circle, then PQ
is the radius.
(v) Square:
A quadrilateral in which all the four angles are right angles and all the four
sides are equal is called a square. Given figure, PQRS is a square.
Ex 5.1 Class 9
Maths 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 atleast 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:
Yes, these postulates contain undefined terms such as ‘Point and Line’. Also,
these postulates are consistent because they deal with two different situations
as
(i) says that given two points A and B, there is a point C lying on the line in
between them. Whereas
(ii) says that, given points A and B, you can take point C not lying on the
line through A and B.
No, these postulates do not follow from Euclid’s postulates, however they
follow from the axiom, “Given two distinct points, there is a unique line that
passes through them.”
Ex 5.1 Class 9
Maths Question 4.
If a point C lies between two points A and
B such that AC = BC, then prove that AC =
Solution:
We have,
AC = BC [Given]
∴ AC + AC = BC + AC
[If equals added to equals then wholes are equal]
or 2AC = AB [∵ AC + BC =
AB]
or AC =
Ex 5.1 Class 9
Maths 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:
Let the given line AB is having two mid points ‘C’ and ‘D’.
AC =
and AD =
Subtracting (i) from (ii), we have
AD – AC =
or AD – AC = 0 or CD = 0
∴ C and D coincide.
Thus, every line segment has one and only one mid-point.
Ex 5.1 Class 9
Maths Question 6.
In figure, if AC = BD, then
prove that AB = CD.
Solution:
Given: AC = BD
⇒ AB + BC = BC + CD
Subtracting BC from both sides, we get
AB + BC – BC = BC + CD – BC
[When equals are subtracted from equals, remainders are equal]
⇒ AB = CD
Ex 5.1 Class 9
Maths Question 7.
Why axiom 5, in the list of
Euclid’s axioms, is 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.’
