NCERT Solutions for Class 9 Maths Chapter 5 Exercise 5.1

Learning geometry fundamentals is important for Class 9 Maths students. Chapter 5, 'Introduction to Euclid's Geometry,' is a cornerstone of mathematical reasoning and proof-writing.

The NCERT Solutions for class 9 maths chapter 5 exercise 5.1 provide complete and reliable answers. They help students understand key concepts like axioms and postulates thoroughly. These solutions are vital for achieving high scores in board and school examinations by ensuring conceptual clarity.

What Class 9 Chapter 5 Introduction to Euclid's Geometry exercise 5.1 covers?

This exercise introduces students to the foundational concepts of Euclidean Geometry. It primarily covers the history and basic definitions established by Euclid in 'The Elements.' Key topics include axioms, postulates, and theorems.

Axioms are self-evident truths accepted without proof, applicable across mathematics. Postulates are similar, but specific to geometry.The exercise requires students to distinguish between these concepts clearly. Students learn to apply Euclid's five postulates and seven axioms to simple geometric statements. This chapter builds the crucial logical groundwork for all subsequent geometry chapters.

Introduction to Euclid's Geometry Class 9 Solutions

NCERT Solutions for Class 9 Maths Chapter 5"Introduction to Euclid's Geometry," covers the basic concepts of geometry, including Euclid's definitions, axioms, and postulates. It emphasizes logical reasoning and deduction, teaching students to understand and apply these fundamental principles to solve geometric problems and develop a solid foundation in geometry.

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 Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

Solution: (i) False There can be 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 its 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 the 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 1 st 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 as follows: Plane: Flat surfaces in which geometric figures can be drawn are known as planes. A plane surface is a surface which lies evenly with straight lines on it. Point: A dimensionless dot which is drawn on a plane surface is known as point. A point is that which has no part. Line: A collection of points that has only length and no breadth is known as a line. It can be extended in both directions. A line is breadth-less length. (i) Parallel lines – Parallel lines are those lines which never intersect each other and are always at a constant perpendicular distance between each other.

(i)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. 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 end points, then the line is known as a line segment. A line segment has 2 end points.(iv) Radius of circle – A radius of a circle is the line from any point on the circumference of the circle to the center of the circle.(v) Square – A quadrilateral in which all the four sides are said to be equal, and each of its internal angles is a right angle, is called square.

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 undefined terms. Undefined terms in the postulates are as follows: – There are many points that lie in a plane. But, in the postulates given here, the position of the point C is not given, as of 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 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 the line segment AB. No, they don’t follow from Euclid’s postulates. They follow the axioms.

4. If a 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 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 mid points of AB. Now, ∴ P and Q are 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 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 mid points of AB. Thus, it is proved that every line segment has one and only one mid-point. 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 from 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’.

How to Prepare using NCERT Solutions Class 9 Chapter 5 Ex 5.1?

Effective preparation uses the NCERT Solutions for class 9 maths chapter 5 exercise 5.1 as a diagnostic and learning tool. Students should not just copy the final answers.

First, try solving all questions independently from the textbook. Then, compare your answers with the NCERT Solutions. Focus on understanding the logical reasoning behind each solution's step. This approach builds strong problem-solving skills and ensures mastery over the chapter's theoretical base.

• Understand Definitions First: Memorize and clearly understand Euclid's five postulates and seven axioms before attempting the exercise.

• Practice Writing Proofs: Pay attention to the structure and precise mathematical language used in the provided solutions' proofs. Replicate this structure in your practice sessions.

• Focus on 'Why': Critically ask yourself why a particular axiom or postulate is applied at each step of a proof. This deepens your conceptual knowledge.

• Self-Assess Regularly: After solving, use the solutions to check your mistakes and identify weak areas for focused, targeted revision before the exam.

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