NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s 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 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. Because through
a single point, the infinite number of lines can pass.
(ii) False.
Because through two distinct points, only one straight line can pass.
(iv) True. Because the radii of congruent (equal) circles are always equal.
(v) True.
AB = PQ … (i)
PQ =
XY … (ii)
From equations
(i) and (ii), we get AB = XY.
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) 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
terms such as point, plane and line need to be defined first.
Here, the terms such as line and right angle need to be defined first.
(iii) Line segment: A line segment is a part of line with a definite length. It
has two end points.
In the figure, the definite length between A and B is a line segment and represented by AB.
Here, the term definite length needs to be defined first.(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 adjoining figure, OA is the radius.
Here, the terms point and centre need to be defined first.
(v) Square: A
square is a rectangle having same length and breadth.
Here, the terms length,
breadth and rectangle need to be defined first.
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 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:
Yes, these postulates contain some undefined terms such as ‘point and line’.
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; whereas
(ii) says that the given points A and B, we can take C not lying on the
line through A and B.
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.”
Ex 5.1 Class 9 Maths Question
4.
If a point C
lies between two points A and B such that AC = BC, then proves that AC = ½ 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
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:
Here, C is
the mid-point of line segment AB, such that AC = BC.
Let us
suppose that there are two mid-points C and C′ of AB.
AC = ½ AB
AC′ = ½ AB
⇒ Points C and C′ are identical.
Hence, every line segment has one and only one mid-point.
Ex 5.1 Class 9 Maths Question
6.
In the given
figure, if AC = BD, then prove that AB = CD.
Solution:
We
have AC = BD
⇒ AB + BC = CD + BC
Subtracting BC from both sides, we get
AB + BC – BC = CD + BC – BC
⇒ AB = CD Hence proved.
Ex 5.1 Class 9 Maths 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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