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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
Solution :
1. Definition of Radius of a Circle
The radius of a circle is defined as the fixed length of the line segment that connects the center of the circle to any point on its circumference. Mathematically, if $O$ is the center and $P$ is any point on the boundary of the circle, the segment $\overline{OP}$ represents the radius $r$.
2. Necessary Prerequisite Terms
To define the "radius of a circle" with logical rigor, the following terms must be defined beforehand:
- Point: A location in space having position but no dimension (length, breadth, or thickness). [Per Euclid’s Definition 1].
- Circle: The set of all points in a plane that are at a fixed distance from a fixed point (the center). [Per Euclid’s Definition 15].
- Center of a Circle: The fixed point from which all points on the circumference are equidistant.
- Line Segment: A part of a line bounded by two distinct points.
- Distance: The measure of the length of the shortest path between two points.
3. Logical Definitions of Prerequisite Terms
| Term | Definition |
|---|---|
| Circle | A plane figure bounded by one curved line, and there is a point within the figure such that all straight lines drawn from it to the circumference are equal to one another. |
| Center | The fixed point described in the definition of a circle; the point of origin for all radii. |
| Circumference | The boundary or the set of points forming the perimeter of the circle. |
4. Geometric Representation
5. Analytical Conclusion
The definition of a "radius" relies heavily on the definitions of a point (the center), a plane (the surface on which the circle is drawn), and the circle itself. Without defining these underlying geometric entities—specifically the center and the distance metric—the term "radius" remains ambiguous. By defining the center as the equidistant anchor and the circle as the locus of points, the radius is established as the invariant measure of that locus.
Final Solution: The radius of a circle is the length of the line segment joining the center to any point on the circumference. Pre-defined terms required: Point, Circle, Center, and Line Segment.
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(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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