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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
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
Solution :
Analysis of the Geometric Statement
The provided statement asserts that if line segment $AB = PQ$ and $PQ = XY$, then $AB = XY$. To evaluate the validity of this statement, we must refer to the fundamental axioms defined by Euclid in his work The Elements.
Step 1: Identifying the Applicable Euclidean Axiom
Euclid's First Axiom (Common Notion 1) states: "Things which are equal to the same thing are equal to one another."
In the context of the given variables:
- $AB = PQ$ (Given)
- $XY = PQ$ (Given as $PQ = XY$, which is symmetric)
Step 2: Logical Derivation
Since both $AB$ and $XY$ are equivalent to the common magnitude $PQ$, we apply the principle of transitivity of equality. If $a = b$ and $b = c$, then $a = c$. [Per Euclid’s First Common Notion].
Visual Representation
Step 3: Conclusion
The statement is True. Because the magnitudes $AB$, $PQ$, and $XY$ represent lengths of line segments on a Euclidean plane, they satisfy the transitive property of equality established in the first of Euclid’s Common Notions.
Final Solution: The statement is True, as things which are equal to the same thing are equal to one another [Euclid's First Common Notion].
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.
- 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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