NCERT Class 9 Maths Solutions: Introduction to Euclids Geometry
Question:
The following statement is true or false? Give reason for your answer.
If two circles are equal, then their radii are equal.
A. True
B. False
Concept in a Minute:
Definition of equal circles.
Explanation:
Two circles are defined as equal if they have the same size. The size of a circle is determined by its radius. Therefore, if two circles are equal, their radii must be equal. This is because the radius is the distance from the center of the circle to any point on its circumference, and for two circles to occupy the same space (be equal), this fundamental measure must be the same.
The statement “If two circles are equal, then their radii are equal” is TRUE.
Reason: The definition of equal circles is that they are congruent, meaning they can be superimposed on each other perfectly. This perfect superimposition can only happen if their corresponding linear measurements, like the radius, are identical. If the radii were different, one circle would be larger or smaller than the other, and they would not be considered equal.
The final answer is $\boxed{A}$.
Question:
The following statement is true or false? Give reason for your answer.
Only one line can pass through a single point.
A. True
B. False
Concept in a Minute:
This question relates to the fundamental postulates of Euclidean geometry concerning points and lines. Specifically, it touches upon the idea of the number of lines that can be drawn through a given point.
Explanation:
The statement “Only one line can pass through a single point” is False. In Euclidean geometry, it is a well-established postulate that through any two distinct points, there is exactly one straight line. However, through a single point, an infinite number of lines can be drawn. Imagine a point on a piece of paper. You can draw a line passing through it going in any direction you choose – upwards, downwards, left, right, or at any angle in between. Each of these directions represents a unique line passing through that single point.
Therefore, the correct answer is False.
Question:
The following statement is true or false? Give reason for your answer.
A terminated line can be produced indefinitely on both the sides.
A. True
B. False
Concept in a Minute:
The question is about the definition of a terminated line in geometry. A terminated line, also known as a line segment, has a definite beginning and end point. In contrast, a line in geometry is defined as extending infinitely in both directions.
Explanation:
A terminated line, by definition, has two endpoints and therefore cannot be produced indefinitely on both sides. It is a finite part of a line. A line, on the other hand, is an infinite, one-dimensional geometric object that extends endlessly in both directions. Therefore, the statement “A terminated line can be produced indefinitely on both the sides” is false.
Answer: B. False
Question:
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
perpendicular lines
Concept in a Minute:
The core concept is understanding the geometric definition of perpendicular lines, which relates to the angle they form when they intersect. This requires a foundational understanding of angles and intersecting lines.
Explanation:
Perpendicular lines are two lines that intersect at a right angle. A right angle measures exactly 90 degrees.
Other terms that need to be defined first:
1. Lines:
Definition: A line is a straight, one-dimensional figure that has no thickness and extends infinitely in both directions. It is the shortest distance between two points.
2. Intersection (of lines):
Definition: The intersection of two lines is the point where they cross or meet.
3. Angle:
Definition: An angle is formed by two rays (or lines) that share a common endpoint, called the vertex. The angle measures the amount of rotation between the two rays.
4. Right Angle:
Definition: A right angle is an angle that measures exactly 90 degrees. It is often indicated by a small square at the vertex where the two lines meet.
With these definitions in place, the concept of perpendicular lines becomes clear: they are lines that intersect to form a 90-degree angle.
Question:
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
parallel lines
Concept in a Minute:
The core concept required is understanding the geometric properties of lines, specifically their relationship in terms of direction and intersection. To define parallel lines, one needs to have a foundational understanding of what a line is and how to describe its orientation.
Explanation:
Parallel lines are defined as two or more lines in the same plane that never intersect, no matter how far they are extended. This means they maintain a constant distance from each other.
Terms that need to be defined first:
1. Line: A line is a one-dimensional geometric object that is straight, infinitely long, and has no width. It can be thought of as a set of points extending infinitely in opposite directions.
2. Plane: A plane is a flat, two-dimensional surface that extends infinitely in all directions. It has length and width but no thickness. Think of a perfectly flat sheet of paper that goes on forever.
3. Intersect: Two or more geometric objects intersect if they have at least one point in common. For lines, intersection means they cross each other at a single point.
Question:
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
radius of a circle
Concept in a Minute:
To understand the radius of a circle, one needs to understand the fundamental components of a circle itself. These include the concept of a point, a distance, and the idea of a collection of points equidistant from a central point.
Explanation:
The radius of a circle is defined as the distance from the center of the circle to any point on its circumference.
Other terms that need to be defined first are:
1. Point: A point is a location in space that has no size, shape, or dimension. It is represented by a dot.
2. Distance: Distance is a measure of the separation between two points. It is always a non-negative value.
3. Circle: A circle is a set of all points in a plane that are at a fixed distance from a given point.
4. Center of a circle: The center of a circle is the fixed point from which all points on the circumference are equidistant.
5. Circumference of a circle: The circumference of a circle is the boundary of the circle, which is the set of all points that are at the fixed distance from the center.
Question:
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.)
Concept in a Minute:
Understanding the nature of axioms and universal truths in mathematics. Axioms are self-evident truths that are accepted without proof. Universal truths are statements that are always true, regardless of context or specific conditions.
Explanation:
Axiom 5 in Euclid’s list states: “The whole is greater than the part.” This is considered a ‘universal truth’ because it is a statement that is self-evidently true and applies to any whole and any of its constituent parts, not just in geometry but in any context where the concept of “whole” and “part” exists. For instance, if you have a whole pizza, any single slice (a part) is clearly smaller than the entire pizza (the whole). This fundamental relationship between a whole and its parts is intuitively understood and universally observed, making it a foundational truth that requires no further proof and holds across all disciplines. Unlike geometric postulates which are specific to geometry, Axiom 5 is a more general logical or philosophical truth.
Question:
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
line segment
Concept in a Minute:
Geometry basics, points, lines, line segments, rays. Understanding fundamental geometric objects and their relationships.
Explanation:
A line segment is a part of a line that has two endpoints.
To define a line segment, we first need to define a point and a line.
Point: A point is a basic geometric element that has no dimension (no length, no width, no thickness). It is often represented by a dot and is used to indicate a specific location.
Line: A line is a straight, one-dimensional figure that extends infinitely in both directions. It has no endpoints. A line can be defined by two distinct points.
Now, we can define a line segment:
Line segment: A line segment is a finite portion of a line that is bounded by two distinct endpoints. It contains all the points on the line between these two endpoints. A line segment can be named by its two endpoints, for example, segment AB or $\overline{AB}$.
Question:
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
square
Concept in a Minute:
The question asks for a definition of the term “square.” To define a square, we need to understand fundamental geometric concepts like “shape,” “polygon,” “quadrilateral,” and “rectangle.”
Explanation:
To define a “square,” we first need to understand these related terms:
Shape: A shape is the outline or form of an object. It describes its appearance in terms of dimensions and boundaries.
Polygon: A polygon is a closed, two-dimensional figure made up of straight line segments.
Quadrilateral: A quadrilateral is a polygon with four sides.
Rectangle: A rectangle is a quadrilateral with four right angles. Its opposite sides are equal in length and parallel.
Now, we can define a square:
Square: A square is a special type of quadrilateral. It is a rectangle where all four sides are of equal length. Equivalently, a square is a quadrilateral with four equal sides and four right angles.
Next Chapter: Linear Equations in Two Variables
Refer Introduction to Euclids Geometry Notes
Practice Introduction to Euclids Geometry Extra Questions
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