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Q1(iii):
Which of the following statements are true and which are false? Give reasons for your answers. (iii) A terminated line can be produced indefinitely on both the sides.
Solution :
Step 1: Theoretical Analysis of the Statement
The statement "A terminated line can be produced indefinitely on both the sides" pertains to the fundamental definitions provided in Euclid’s Elements, Book I. In modern geometry, a "terminated line" corresponds to what is formally defined as a line segment. The act of "producing" it indefinitely refers to extending the endpoints of the segment along the same path to form a straight line.
Step 2: Justification via Euclid’s Postulates
This statement is a direct paraphrase of Euclid’s Second Postulate. [Per Euclid’s Second Postulate: "A terminated line can be produced indefinitely."] Euclid defined a "terminated line" as a line segment with two fixed endpoints. His postulate asserts that these endpoints do not restrict the length of the line in a mathematical sense; rather, the segment exists as part of an infinite linear continuum.
Step 3: Visual Representation of the Concept
Step 4: Analytical Conclusion
- Status: True.
- Reasoning: By the definitions of Euclidean geometry, a line is defined as having length without breadth and exists infinitely. A segment ($AB$) can be extended in either direction to satisfy the properties of an infinite straight line, adhering to the axiomatic structure established in the Elements.
Final Solution: The statement is True. It is consistent with Euclid’s Second Postulate, which affirms that a finite line segment (terminated line) may be extended infinitely to form a straight line.
More Questions from Class 9 Mathematics Introduction to Euclid’s Geometry EXERCISE 5.1
- Q1(i): 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.
- Q1(ii): Which of the following statements are true and which are false? Give reasons for your answers. (ii) There are an infinite number of lines which pass through two distinct points.
- Q1(iv): Which of the following statements are true and which are false? Give reasons for your answers. (iv) If two circles are equal, then their radii are equal.
- Q1(v): Which of the following statements are true and which are false? Give reasons for your answers. (v) In Fig. 5.9, if $AB = PQ$ and $PQ = XY$, then $AB = X
- Q2(i): 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
- Q2(ii): 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? (ii) perpendicular lines
- Q2(iii): 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? (iii) line segment
- Q2(iv): 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? (iv) radius of a circle
- Q2(v): 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? (v) square
- Q3: 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.
- Q4: If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$. Explain by drawing the figure.
- Q5: 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.
- Q6: In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
- Q7: 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.)
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