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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.
Solution :
Step 1: Analyzing the Geometric Axioms
To determine the veracity of the statement, we must evaluate it against Euclid’s first postulate, which serves as the foundational axiom for the existence of lines in Euclidean geometry.
- Euclid’s First Postulate: "A straight line may be drawn from any one point to any other point." [Per Euclid’s Elements, Book I, Postulate 1].
- Euclid’s First Axiom (Implicit): Given two distinct points $P$ and $Q$, there exists one and only one straight line that passes through both points. [Per Euclid’s Elements, Book I, Postulate 2, implying uniqueness].
Step 2: Logical Derivation & Geometric Proof
Let there be two distinct points $A$ and $B$ in a plane. A line is defined as the unique shortest distance between two points. If we assume there are two lines, $l_1$ and $l_2$, passing through points $A$ and $B$, we encounter a contradiction with the property of uniqueness in Euclidean space.
Step 3: Evaluating the Statement
| Statement | Verdict | Reasoning |
|---|---|---|
| There are an infinite number of lines which pass through two distinct points. | False | According to Euclid’s axioms, there is one and only one unique straight line that passes through two distinct points. |
Step 4: Conclusion
The statement contradicts the fundamental theorem derived from Euclid’s postulates, which states that two distinct points determine a unique line. While there are an infinite number of lines that can pass through a single point, the addition of a second distinct point restricts the possibility to exactly one line.
Final Solution: The statement is False. Per Euclid’s axioms, one and only one unique straight line can pass through two distinct points.
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(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.
- 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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