NCERT Solutions For Class 9 Maths Chapter 5 Introduction To Euclid’s Geometry Exercise 5.1 – 2025-26
NCERT Solutions for Class 9 Maths Chapter 5 Exercise 5.1 – Introduction to Euclid’s Geometry by Mathify.in
Mathify.in provides a comprehensive set of solutions for Exercise 5.1 in Class 9 Maths Chapter 5, Introduction to Euclid’s Geometry. This exercise serves as a crucial stepping stone in understanding key concepts of Euclidean geometry. It helps students identify and interpret geometric shapes, recognize points, lines, and planes in diagrams, and assess the truth value of geometric statements—such as the idea that only one line can pass through a single point.
By working through these problems, learners build a strong foundation for exploring the properties and proofs that define classical geometry. The solutions offered by Mathify.in are designed to clarify these principles and support confident problem-solving.
NCERT Solutions for Class 9 Maths Chapter 5 Exercise 5.1 – by Mathify.in
This exercise centers on essential concepts in Euclidean geometry. With Mathify.in’s Class 9 Maths NCERT Solutions, students receive detailed, step-by-step explanations for every question in Exercise 5.1—ensuring a strong grasp of foundational ideas.
Key concepts covered include:
- 🔹 Point: A location in space with no dimensions—no length, width, or thickness. Examples: A, B, C.
- 🔹 Line: A breadthless length that extends infinitely in both directions.
- 🔹 Line Segment: A portion of a line that has two defined endpoints.
- 🔹 Ray: A part of a line that starts at one point and extends infinitely in one direction.
- 🔹 Plane: A flat, two-dimensional surface that stretches infinitely in all directions.
Basic Properties Introduced:
- A straight line segment can be drawn joining any two points.
- A terminated line can be extended indefinitely.
Exercise 5.1 contains 7 fully solved questions, and Mathify.in ensures that learners understand these basics thoroughly before progressing to advanced topics.
Solution:-
(i) Only one line can pass through a Single Point.
Ans: “An infinite number of lines can pass through a single point P, as illustrated in the diagram. Each line represents a unique direction, demonstrating that countless lines can intersect at one location.”
(ii) There are an infinite number of lines which pass through two distinct points.
Ans: False
“Only one unique straight line can be drawn through two distinct points, P and Q, as depicted in the diagram.”
(iii) A terminated line can be produced indefinitely on both the sides.
Ans: True
- “A terminated line can be extended infinitely in both directions.”
- “Let AB be a terminated line segment. It is evident that this line can be extended infinitely in both directions.”
(iv) If two circles are equal, then their radii are equal.
Ans: True
“If two circles are equal, their centers and circumferences are the same, and so are their radii.”
(v) In the following figure, if AM = PQ and PQ = XY, then AB = XY
Ans: True
Application of Euclid’s First Axiom
Let AB and XY be two line segments (terminated lines), and let PQ be a third line segment.
It is given that:
AB = PQ and XY = PQ
According to Euclid’s First Axiom:
“Things which are equal to the same thing are equal to one another.”
Applying this axiom, since both AB and XY are equal to PQ, it follows that: AB = XY
Solution:-
Point:
A point is represented by a small dot made using a sharp pencil on paper. It has no length, breadth, or height—meaning it has no dimensions. A point only indicates a specific location or position in space.
Line:
A line can be illustrated by folding a piece of paper, stretching a string tightly, or observing the edge of a ruler. These are all examples of geometrical lines. Fundamentally, a line is straight and extends endlessly in both directions.
Plane:
A plane can be visualized as a flat surface like the smooth face of a wall or the surface of a sheet of paper. These everyday examples help illustrate the idea of a geometrical plane, which extends endlessly in all directions without any thickness.
Ray:
A ray is a part of a line that starts from a fixed point and extends endlessly in one direction. It has one endpoint and continues in the direction of another point on the line.
Angle:
An angle is formed when two rays originate from the same point but are not aligned in the same direction. These two non-collinear rays share a common starting point called the vertex.
Circle:
A circle is a set of all points in a plane that are equidistant from a fixed point. This fixed point is known as the centre of the circle.
Quadrilateral:
A quadrilateral is a closed figure formed by joining four line segments in such a way that no three segments are collinear.
(i) Parallel Lines:
Parallel lines are straight lines that lie in the same plane and never intersect, no matter how far they are extended. The perpendicular distance between them remains constant at every point.
To understand the concept of parallel lines, we must be familiar with points, lines, and the uniform distance between lines that do not meet or intersect.
(ii) Perpendicular Lines:
Perpendicular lines are two lines that intersect at a point and form a right angle (90°) with each other.
To understand perpendicular lines, it’s important to first grasp the concepts of a line and an angle. A line extends infinitely in both directions, and an angle is formed when two rays meet at a common endpoint.
(iii) Line Segment:
A line segment is a straight path that connects two distinct points. Unlike a line, it does not extend indefinitely—it has a definite beginning and end.
To define a line segment properly, one must first understand the concept of a point and how a straight line is formed between two such points.
(iv) Radius of a Circle:
The radius of a circle is the fixed distance from its centre to any point located on the circle.
To define the radius accurately, it is essential to first understand the concepts of a point and a circle. A point represents a position in space, and a circle is a set of points that are all at the same distance from a fixed centre.
(v) Square:
A square is a special type of quadrilateral in which all four sides are equal in length and each angle measures exactly 90 degrees.
To understand a square properly, one must first be familiar with the concepts of a quadrilateral, equal sides, and right angles.
Solution:-
The given postulates contain several undefined terms, which are foundational to the development of geometric reasoning. Since these postulates refer to distinct scenarios, they are logically consistent with one another.
Moreover, any conclusion that contradicts an established axiom or postulate cannot be logically derived. Specifically, the outcomes discussed here do not arise directly from Euclid’s postulates but instead follow from the axiom: “Given two distinct points, there is a unique line that passes through them.”
Solution:-
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Axiom 5: The Whole is Greater Than the Part
This axiom states that the whole is always greater than any of its individual parts. It is considered a universal truth because it applies not only in mathematics but across various real-life situations.
Let’s understand this axiom with the help of two examples—one mathematical and one from everyday life:
Case 1: Mathematical Example
Let t represent a whole quantity, and let a, b, and c be its parts.
So,
t = a + b + c
Here, it is evident that the total quantity t is greater than each of its parts (a, b, or c) individually.
Hence, the axiom “The whole is greater than the part” holds true.
Case 2: Real-Life Example
Consider the continent Asia, and the country India, which is a part of Asia.
Although India lies within Asia, Asia as a whole is larger than India in terms of area and scale.
Therefore, Asia (the whole) is greater than India (the part).
This demonstrates that the axiom is universally valid—not just in mathematics but in real-world contexts as well.