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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
Solution :
Definition of Line Segment
A line segment is defined as a part of a line that is bounded by two distinct endpoints. It consists of the set of all points on a line that lie between and include two specified points, $A$ and $B$. [Per the Euclidean conceptualization of dimensionality, a line segment represents a one-dimensional figure with a finite, measurable length].
Requirement for Prior Definitions
To rigorously define a "line segment," the following primitive terms (undefined terms in Euclid’s geometry) and secondary terms must be established first:
- Point: That which has no part (zero-dimensional entity).
- Line: Breadthless length (extending infinitely in both directions).
- Betweenness: A spatial relation defining the ordering of points on a line.
Defining Necessary Prerequisite Terms
| Term | Definition |
|---|---|
| Point | A location in space having position but no magnitude, dimension, or extent. |
| Line | A one-dimensional geometric object that extends infinitely in two opposite directions and is defined by the locus of points. |
| Betweenness | For any three collinear points $A, B, C$, point $B$ is said to lie between $A$ and $C$ if the distance $AB + BC = AC$. |
Visual Representation of a Line Segment
Logical Derivation
If we denote a line as $L$ and two distinct points $A, B \in L$, then the line segment $\overline{AB}$ is formally defined as the set $\{P \mid P \in L \text{ and } P \text{ is between } A \text{ and } B, \text{ or } P = A, \text{ or } P = B\}$. This satisfies the condition of finite extent, distinguishing it from an infinite line or a ray.
Final Solution: A line segment is a part of a line with two definite endpoints. The definitions of "point," "line," and "betweenness" are prerequisite requirements for this definition.
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(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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