NCERT Solutions for Class 9 Maths Exercise 5.1 Question 3
Understanding the Question 🧐
This question presents two statements, or ‘postulates’, and asks us to analyze them. We need to check three things: Do they use undefined terms? Are they consistent with each other? And do they come from Euclid’s famous postulates? This is a great exercise to understand the logical foundations of geometry. Let’s break it down with these ncert solutions.
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.
Part 1: Analyzing Undefined Terms 📝
Answer: Yes, these postulates contain undefined terms.
The fundamental terms used here are ‘point’ and ‘line’. In Euclidean geometry, these are considered undefined because we cannot define them using simpler terms. We can describe them:
- A point is a location with no size or dimension.
- A line is a straight path that extends infinitely and has no thickness.
We accept their existence and meaning intuitively, and all other geometric definitions are built upon them. So, both postulates rely on these undefined terms.
Part 2: Checking for Consistency 🤔
Answer: Yes, these postulates are consistent.
Consistency means that the statements do not contradict each other. They can both be true in the same system without creating a paradox. Let’s see why they are consistent:
- Postulate (i) describes a situation where points are on the same line (collinear). It says that between any two points on a line, you can find another.
- Postulate (ii) describes a situation where points are not on the same line (non-collinear). It says that not all points in existence lie on a single straight line.
These two statements deal with two completely different scenarios. One talks about what happens *on* a line, and the other talks about what happens *off* a line. Since they don’t make opposing claims about the same situation, they are perfectly consistent.
Part 3: Relationship with Euclid’s Postulates 📜
Answer: No, these postulates do not follow directly from Euclid’s postulates.
While these statements are true in Euclidean geometry, they are considered independent axioms that are consistent with Euclid’s work, but not derived from his original five postulates. Here’s why:
- Regarding Postulate (i): Euclid’s postulate 2 says a line segment can be extended indefinitely. It does not explicitly state that a line is made of infinitely many points, or that a point always exists between any two distinct points. This idea is an axiom (often attributed to Hilbert) that makes the line “dense.”
- Regarding Postulate (ii): Euclid’s postulates allow us to construct lines and figures, but they don’t explicitly state that a plane must exist (i.e., that there are points not on a single line). This second postulate ensures that our geometry is at least two-dimensional and not just a single line. It’s a foundational assumption for plane geometry.
In summary: These postulates are not theorems that can be proven from Euclid’s five postulates. Instead, they are separate axioms that are typically added to create a more rigorous foundation for geometry.
Conclusion and Key Points ✅
To sum up our analysis:
- The given postulates contain undefined terms like ‘point’ and ‘line’.
- They are consistent because they describe different situations that don’t contradict each other.
- They do not follow from Euclid’s postulates but are independent axioms that are compatible with Euclidean geometry.
- Undefined Terms: ‘Point’ and ‘Line’ are the basic building blocks of geometry.
- Consistency: A set of rules is consistent if no rule contradicts another.
- Axioms vs. Postulates: Both are assumptions. Euclid’s five postulates were specific to geometry, while these two statements are considered broader axioms in modern geometry.
FAQ
Q: What are the main undefined terms in geometry?
A: The main undefined terms in geometry are ‘point’ and ‘line’. We understand them intuitively as a location and a straight path, but we don’t define them using simpler terms.
Q: What does it mean for postulates to be ‘consistent’?
A: A set of postulates is consistent if it’s impossible to deduce a contradiction from them. They should not lead to statements that are logically opposite. They represent different facts about the same system that can all be true at once.
Q: Does the first given postulate (a point &&C&& is between &&A&& and &&B&&) follow from Euclid’s postulates?
A: No, it does not directly follow from Euclid’s postulates. Euclid’s work implies a line can be extended, but it doesn’t explicitly state that a line contains an infinite number of points. This statement is considered a separate axiom in modern geometry.
Q: How is the second given postulate (at least three non-collinear points exist) related to Euclid’s work?
A: This also does not follow directly from Euclid’s postulates but is a fundamental axiom. It ensures that geometry is not just a single line but exists in a plane (at least two dimensions). It’s consistent with Euclid’s geometry but not derived from it.
Q: Do the two given postulates in this question contradict each other?
A: No, they do not contradict each other. One postulate deals with points on a single line, and the other deals with points that are not on the same line. They describe two different situations and are therefore consistent.
Further Reading
To deepen your understanding of axioms and postulates, refer to your Class 9 Maths textbook or visit the official NCERT website: https://ncert.nic.in/. These concepts form the logical backbone of all mathematics.
