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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.
Solution :
1. Theoretical Foundation: Defining Equality in Geometry
In Euclidean geometry, two geometric figures are considered equal if they are congruent, meaning they can be superimposed to coincide exactly [Euclid’s Common Notion 4: "Things which coincide with one another are equal to one another"].
2. Geometric Analysis of the Circle
A circle is defined as the locus of all points in a plane that are at a fixed distance, known as the radius ($r$), from a fixed point, known as the center ($O$). The equation of a circle in Cartesian coordinates is given by:
$(x - h)^2 + (y - k)^2 = r^2$
where $(h, k)$ represents the coordinates of the center and $r$ represents the radius [Definition of a Circle].
3. Logical Derivation of Equality
Consider two circles, $C_1$ and $C_2$, with radii $r_1$ and $r_2$ respectively.
- If $C_1$ and $C_2$ are equal, they must coincide when placed one upon the other [Euclid’s Common Notion 4].
- For two circles to coincide, their boundaries (circumferences) must overlap perfectly.
- The boundary of a circle is uniquely determined by the distance of all its points from the center, which is defined as the radius.
- If $r_1 \neq r_2$, the set of points defined by the two circles will occupy different areas in the plane and cannot coincide.
- Therefore, the condition of equality necessitates that $r_1 = r_2$.
4. Statement Verification
Statement: "If two circles are equal, then their radii are equal."
Result: True.
Reasoning: By the principle of superimposition, since the radius is the constant distance from the center that defines the extent of the circle, two circles can only occupy the same space (be equal) if their constant distances from the center are identical.
Final Solution: The statement is True. By the definition of a circle and Euclid’s Common Notion 4, congruent circles must have identical radii, as the radius is the fundamental parameter determining the circle's spatial extent.
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(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.)
CBSE Solutions for Class 9 Mathematics Introduction to Euclid’s Geometry
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