NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclids Geometry – We study about Euclid’s approach to geometry and tries to link it with the present-day geometry. Introduction to Euclid’s Geometry provides the students with a method of defining common geometrical shapes and terms.
ALSO CHECK – NCERT Solutions for Class 10 Maths
NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclids Geometry is prepared by our best subject experts teachers group thats help students to understand all the topics easily. These Solutions of NCERT Maths help the students in solving the problems efficiently for the upcoming exams. With the help of these NCERT Solutions for Class 9 Maths, students can understand the complex topics of class 9 Maths.
ALSO CHECK – Download the free Class 10th Notes here
NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid 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 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: A line that is terminated can be indefinitely produced on both sides as a line can be extended on both its sides infinitely. Hence, the statement mentioned is True.
(iv) True
Reason : The radii of two circles are equal when the two circles are equal. The circumference and the centre of both the circles coincide; and thus, the radius of the two circles should be equal. Hence, the statement mentioned is True.
(v) True
Reason : According to Euclid’s 1st axiom- “Things which are equal to the same thing are also equal to one another”. Hence, the statement mentioned is True.
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 𝐴𝐵⎯⎯⎯⎯⎯⎯⎯⎯ or 𝐵𝐴⎯⎯⎯⎯⎯⎯⎯⎯.
(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.
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.”
Question 4.
If a point C lies between two points A and B such that AC = BC, then prove that AC = 12 AB, explain by drawing the figure.
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 = 12𝐴𝐵
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 = 12𝐴𝐵 ……(i)
and AD = 12𝐴𝐵 ……(ii)
Subtracting (i) from (ii), we have
AD – AC = 12𝐴𝐵−12𝐴𝐵
or AD – AC = 0 or CD = 0
∴ C and D coincide.
Thus, every line segment has one and only one mid-point.
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
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:
The whole is always greater than the part.
For Example: A cake. When it is whole or complete, assume that it measures 2 pounds but when a part from it is taken out and measured, its weigh will be smaller than the previous measurement. So, the fifth axiom of Euclid is true for all the materials in the universe. Hence, Axiom 5, in the list of Euclid’s axioms, is considered a ‘universal truth.
NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid Geometry Exercise 5.2
Question 1.
How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
Solution:
We can write Euclid’s fifth postulate as ‘Two distinct intersecting lines cannot be parallel to the same line.’
Question 2.
Does Euclid’s fifth postulate imply the existence of parallel lines ? Explain.
Solution:
Yes. If a straight line l falls on two lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate, lines m and n will not meet on this side of l. Also, we know that the sum of the interior angles on the other side of the line l will be two right angles too. Thus, they will not meet on the other side also.
∴ The lines m and n never meet, i.e, They are parallel.