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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
Solution :
Step 1: Defining Perpendicular Lines
Two lines, $l$ and $m$, are said to be perpendicular if they intersect at a point such that the angle formed between them is exactly $90^\circ$ (a right angle). In Euclidean geometry, if a line $AB$ is perpendicular to line $CD$, the intersection forms four right angles [Per Euclid’s Definition 10].
Step 2: Identifying Preliminary Terms
To define "perpendicular lines" with mathematical rigor, the following terms must be defined first:
- Line: A breadthless length [Per Euclid’s Definition 2].
- Intersection: The condition where two geometric figures share at least one common point.
- Angle: The inclination of two straight lines to one another meeting at a point [Per Euclid’s Definition 8].
- Right Angle: When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle [Per Euclid’s Definition 10].
- Degree ($^\circ$): A unit of measurement for angles where a full rotation is $360^\circ$ and a right angle is defined as $90^\circ$.
Step 3: Formal Definitions of Preliminary Terms
| Term | Definition |
|---|---|
| Line | A one-dimensional geometric object that extends infinitely in both directions. |
| Angle | The space (measured in degrees or radians) between two intersecting lines. |
| Right Angle | An angle measuring exactly $90^\circ$, representing one-quarter of a full revolution. |
Step 4: Geometric Visualization
Step 5: Logical Synthesis
By defining "Line" as an infinite set of points, "Angle" as the rotation between two rays originating from a common vertex, and "Right Angle" as a specific rotational value of $90^\circ$, we establish the necessary axiomatic foundation [Per Euclid’s Postulates]. Thus, perpendicularity is the state of incidence where the angular value between two lines is exactly one right angle.
Final Solution: Perpendicular lines are two lines that intersect at a common point such that the angle between them is $90^\circ$. The necessary preliminary terms defined are Line, Intersection, Angle, and Right Angle.
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(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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