Understanding Euclidian Geometry is easy if students refer to NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry Exercise 5.1. This exercise are designed to help them grasp the conceptual basis and solve numerical problems. Traditionally, Euclidian Geometry is an elementary concept, but it has significance in various other fields of study. Therefore, it is crucial to have a good grasp of this concept from the Class 9 Maths NCERT Solutions Chapter 5.
Students will find that the NCERT solution is one of the best resources to learn. This is due to the fact that the solutions are designed by a team of dedicated teachers with many years of experience. Consequently, NCERT Solutions is one of the best guides that students might come across for their studies. Important topics are presented in a simple, structured format. Moreover, complex jargon is broken down or explained in the content. Furthermore, content is refreshed regularly as per the prescribed CBSE syllabus.
NCERT Solutions for Class 9 Maths Chapter 5 – Introduction to Euclid’s Geometry Exercise 5.1
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Exercise 5.2 Solutions – 2 Questions
NCERT Solutions for Class 9 Maths Chapter 5 – Introduction to Euclid’s Geometry Exercise 5.1
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 sides.
(iv) If two circles are equal, then their radii are equal.
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
Solution:
(i) False
There can be an infinite number of lines that can be drawn through a single point. Hence, the statement mentioned is False.
(ii) False
Through two distinct points, there can be only one line that can be drawn. Hence, the statement mentioned is False.
(iii) True
A line that is terminated can be indefinitely produced on both sides, as a line can be extended on both sides infinitely. Hence, the statement mentioned is True.
(iv) True
The radii of two circles are equal when the two circles are equal. The circumference and the centre of both circles coincide; and thus, the radius of the two circles should be equal. Hence, the statement mentioned is True.
(v) True
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.
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, there are other terms which need to be defined first; they are
Plane: Flat surfaces in which geometric figures can be drawn are known are planes. A plane surface is a surface which lies evenly with straight lines on itself.
Point: A dimensionless dot which is drawn on a plane surface is known as a point. The point is that which has no part.
Line: A collection of points that has only length and no breadth is known as a line. And it can be extended in both directions. A line is a breadthless length.
(i) Parallel lines – Parallel lines are those lines which never intersect each other and are always at a constant distance perpendicular to each other. Parallel lines can be two or more lines.
(ii) Perpendicular lines – Perpendicular lines are those lines which intersect each other in a plane at right angles, and then the lines are said to be perpendicular to each other.
(iii) Line Segment – When a line cannot be extended any further because of its two endpoints, then the line is known as a line segment. A line segment has 2 endpoints.
(iv) Radius of a circle – A radius of a circle is the line from any point on the circumference of the circle to the centre of the circle.
(v) Square – A quadrilateral in which all four sides are said to be equal is called a square, and each of its internal angles is a right angle.
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 Euclid’s postulates? Explain.
Solution:
Yes, these postulates contain undefined terms. Undefined terms in the postulates are
– There are many points that lie in a plane. But, in the postulates given here, the position of point C is not given as to whether it lies on the line segment joining AB or not.
– On top of that, there is no information about whether the points are in the same plane or not.
And
Yes, these postulates are consistent when we deal with these two situations:
– Point C is lying on the line segment AB in between A and B.
– Point C does not lie on line segment AB.
No, they don’t follow Euclid’s postulates. They follow the axioms.
4. If point C lies between two points A and B, such that AC = BC, then prove that AC = ½ AB. Explain by drawing the figure.
Solution:
Given that, AC = BC
Now, adding AC on both sides,
L.H.S.+AC = R.H.S.+AC
AC+AC = BC+AC
2AC = BC+AC
We know that BC+AC = AB (as it coincides with line segment AB)
∴ 2 AC = AB (If equals are added to equals, the wholes are equal.)
⇒ AC = (½)AB
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 AB be the line segment.
Assume that points P and Q are the two different midpoints of AB.
Now,
∴ P and Q are the midpoints of AB.
Therefore,
AP = PB and AQ = QB
also,
PB+AP = AB (as it coincides with line segment AB)
Similarly, QB+AQ = AB
Now,
Adding AP to the L.H.S. and R.H.S. of the equation AP = PB
We get AP+AP = PB+AP (If equals are added to equals, the wholes are equal.)
⇒ 2AP = AB — (i)
Similarly,
2 AQ = AB — (ii)
From (i) and (ii), Since R.H.S. are the same, we equate the L.H.S.
2 AP = 2 AQ (Things which are equal to the same thing are equal to one another.)
⇒ AP = AQ (Things which are double of the same things are equal to one another.)
Thus, we conclude that P and Q are the same points.
This contradicts our assumption that P and Q are two different midpoints of AB.
Thus, it is proved that every line segment has one and only one midpoint.
Hence, proved.
6. In Fig. 5.10, if AC = BD, then prove that AB = CD.
Solution:
It is given AC = BD
From the given figure, we get
AC = AB+BC
BD = BC+CD
⇒ AB+BC = BC+CD [AC = BD, given]
We know that, according to Euclid’s axiom, when equals are subtracted from equals, remainders are also equal.
Subtracting BC from the L.H.S. and R.H.S. of the equation AB+BC = BC+CD, we get
AB+BC-BC = BC+CD-BC
AB = CD
Hence, proved.
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:
Axiom 5: 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 of it is taken out and measured, its weight 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.
Euclid’s Geometry is a system introduced by Euclid, an Alexandrian-Greek Mathematician, way back in 300 BC. 2,000 years later, Euclid’s contributions still remain valid, and it has been adopted in various applications, ranging from engineering to theoretical physics. Euclidian Geometry also has significant academic implications in the disciplines of science and mathematics. Find out how Euclidean Geometry works and explore the 2000-year-old theorems. Also, check out other important NCERT Solutions for Class 9 Maths to help students practise various other mathematical concepts.
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