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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.
Solution :
1. Analysis of Undefined Terms
In the field of axiomatic geometry, certain terms are accepted as primitive (undefined) to prevent circular reasoning. The two provided postulates contain the following undefined terms:
- Point: A location in space with no dimension.
- Line: A one-dimensional figure extending infinitely in both directions.
- Between/In-between: An ordering relation defining relative position on a line segment.
[By Hilbert's axiomatic system, terms like 'point' and 'line' are fundamental and cannot be defined by simpler geometric concepts without logical circularity.]
2. Consistency Analysis
A set of postulates is consistent if they do not contradict one another. Postulate (i) describes the density of points on a line segment (a property of Euclidean space), while Postulate (ii) describes the existence of a plane (non-collinearity). These two statements are mathematically independent and do not lead to contradictory deductions. Therefore, the postulates are consistent.
3. Alignment with Euclid’s Postulates
We must evaluate these against the original five postulates of Euclid:
| Statement | Relationship to Euclid |
|---|---|
| (i) Existence of a point $C$ between $A$ and $B$. | This is not explicitly stated in Euclid's five postulates. However, it is consistent with Euclid’s Axiom of Betweenness, which is implicitly assumed in his construction of a straight line. |
| (ii) Existence of three non-collinear points. | Euclid assumes the existence of space and planes implicitly. This statement is a requirement to define a plane, which Euclid assumes as a foundational given rather than deriving from his specific five postulates. |
4. Visual Representation of Non-Collinear Points
Conclusion
The postulates are logically sound and consistent within the framework of Euclidean Geometry. While they are not explicitly listed in Euclid’s original five postulates, they do not contradict them; rather, they function as fundamental assumptions necessary to describe the spatial nature of lines and planes that Euclid presupposed in his works. They follow from the implicit geometric intuition that underpins Euclid’s system.
Final Solution: The postulates contain undefined terms ("point", "line", "between"). They are consistent as they do not contradict one another. They do not follow directly from Euclid's five postulates but are consistent with the implicit assumptions of his geometric system.
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(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
- 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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