Introduction To Euclid’s Geometry  Class 9 Mathematics  Chapter 5  Notes, NCERT Solutions & Extra Questions
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Exercise 5.1  Introduction To Euclid’s Geometry  NCERT  Mathematics  Class 9
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.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) if lines $A B=P Q$ and $P Q=X Y$, then $A B=X Y$.
Let's analyze each statement individually:
(i) Only one line can pass through a single point.
 False. Through a single point, an infinite number of lines can pass. The reason is that a line is defined by two distinct points, and for any given point, there can be an infinite number of other points that lie in different directions from the initial point, each pair defining a distinct line.
(ii) There are an infinite number of lines which pass through two distinct points.
 False. There is exactly one unique line that can pass through any two distinct points, according to one of the fundamental axioms of geometry. The uniqueness of this line is what allows us to define it by just those two points.
(iii) A terminated line can be produced indefinitely on both the sides.
 True. This statement is essentially describing a line segment and its property of being extendable to form a line of infinite length. A line segment, which appears "terminated" with two endpoints, can indeed be extended indefinitely in both directions to form a line. This is based on the principle of extension in geometry.
(iv) If two circles are equal, then their radii are equal.
 True. For two circles to be considered equal, all their dimensions and properties must match exactly. Since the radius is fundamental to a circle's properties (defining its size), equal circles must have equal radii. This is based on the definition of circle equality.
(v) if lines (AB=PQ) and (PQ=XY), then (AB=XY).
 True. This statement demonstrates the transitive property of equality applied to the lengths of line segments. If line segment (AB) is equal in length to line segment (PQ), and (PQ) is equal in length to (XY), then it follows that (AB) is equal in length to (XY).
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
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square
To understand the concepts requested, it's helpful to first define some fundamental geometric terms:
 Line: An infinitely long, straight figure with no thickness, extending in both directions.
 Point: A specific location in space with no size, dimension, or thickness.
 Plane: A flat, twodimensional surface that extends infinitely in all directions.
Now, let's define the requested terms:
(i) Parallel lines: Two lines in the same plane that do not intersect, no matter how far they are extended. They remain the same distance apart over their entire length.
(ii) Perpendicular lines: Two lines that intersect at a right angle (90 degrees). When two lines are perpendicular, they form four right angles at the point of intersection.
(iii) Line Segment: A part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
(iv) Radius of a Circle: The distance from the center of the circle to any point on its circumference. It is also half the length of the circle's diameter.
(v) Square: A foursided polygon (quadrilateral) that has all sides of equal length and all angles equal to 90 degrees. A square is both a rhombus (all sides equal in length) and a rectangle (all angles equal to 90 degrees), making it a regular quadrilateral.
These definitions build upon foundational concepts such as lines, points, and planes to describe more complex geometric figures and relationships.
Consider two 'postulates' given below:
(i) Given any two distinct points $\mathrm{A}$ and $\mathrm{B}$, there exists a third point $\mathrm{C}$ which is in between $\mathrm{A}$ and $\mathrm{B}$.
(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.
The two postulates you've presented use the terms "points" and "line", as well as the concept of "in between" and "on the same line". Here's a breakdown:

Undefined Terms:
 The terms "point" and "line" are indeed undefined within the context of these postulates. In Euclidean geometry, and more broadly in the axiomatic formulation of geometry itself, both points and lines are considered primitive notions. This means they are not defined in terms of more basic concepts, but their properties and interrelations are described by axioms or postulates.
 The concept of "in between" for the case of point $\mathrm{C}$ being between points $\mathrm{A}$ and $\mathrm{B}$ is also an undefined term within these postulates. It's a geometric intuition that is understood rather than explicitly defined in terms of more fundamental notions.
 The phrase "on the same line" refers to collinearity, which, while not an undefined term itself, relies on the primitive notions of "point" and "line" without providing a specific definition within the context of these postulates.

Consistency:
 These postulates appear to be consistent with each other as there's no inherent contradiction between them. The first postulate speaks to the denseness of points on a line segment, while the second establishes the existence of a plane by asserting the existence of three noncollinear points. They can be used in conjunction with other axioms or postulates to build a coherent geometric system.

Relation to Euclid's Postulates:
 The first postulate shares an intuitive agreement with the idea implicit in Euclid's first postulate, which states that a straight line can be drawn from any point to any other point. However, it introduces a notion of denseness (that between any two points, another can be found) which isn't directly mentioned by Euclid.
 The second postulate is somewhat reflected in Euclid's axioms through his postulate that asserts the possibility to draw a circle with any center and radius, implying the existence of an infinite number of points not all on the same line. However, the specific claim about three noncollinear points is not explicitly made in Euclid's postulates but is necessary for defining a plane in Euclidean geometry.
In conclusion, the discussed postulates contain undefined terms fundamental to geometrical axioms, are consistent with each other, and while they are not directly stated in Euclid’s postulates, they are compatible with the principles of Euclidean geometry. These postulates could be viewed as supplementary to Euclid's to foster a deeper understanding of geometrical concepts.
If a point $\mathrm{C}$ lies between two points $\mathrm{A}$ and $\mathrm{B}$ such that $\mathrm{AC}=\mathrm{BC}$, then prove that $\mathrm{AC}=\frac{1}{2} \mathrm{AB}$. Explain by drawing the figure.
point $C$ is called a midpoint of line segment $A B$. Prove that every line segment has one and only one midpoint.
Proof for AC = 1/2 AB when C is equidistant from A and B:
 Given: C is a point on line segment AB such that AC = BC.
 To Prove: AC = 1/2 AB.

Proof:
 Since C lies between A and B, AC + CB = AB (by the Segment Addition Postulate).
 Given that AC = BC, let's substitute BC for AC in the equation: AC + AC = AB or, 2AC = AB.
 Dividing both sides by 2 gives us: AC = 1/2 AB.
Drawing the figure:
Let's start by drawing the figure. A, B, and C are points on a line where C lies between A and B and AC = BC.
Unfortunately, I can't generate or display images directly, but I can describe how to draw it:
 Draw a horizontal line.
 Mark points A and B on the line such that B is to the right of A.
 Find and mark point C exactly between A and B.
 Label lengths AC and BC to show they are equal.
Proving every line segment has one and only one midpoint:
 Given: A line segment AB.
 To Prove: There exists exactly one point C on AB such that AC = BC.

Proof of Existence:
 Consider the bisector of the line segment AB. By the definition of a bisector, it divides AB into two equal parts.
 Let C be the point of intersection of the line segment AB with the bisector. By the definition of a bisector, AC = BC.

Proof of Uniqueness:
 Suppose for contradiction there exists another point D (not equal to C) on AB such that AD = BD.
 Without loss of generality, assume D lies between A and C. Then, AD < AC because D is closer to A than C is.
 But we also said that AD = BD, and since AC = BC, this would imply AD < BD, a contradiction since they are supposed to be equal.
 Hence, there cannot exist another point like D that divides AB into two equal segments, proving the uniqueness.
Together, these parts show that for any line segment AB, there exists exactly one midpoint C that divides AB into two equal segments AC and BC.
If ABCD is a straight line, if $A C=B D$, then prove that $A B=C D$.
To prove $AB = CD$ under the condition that $AC = BD$ for a straight line segment ABCD, we can use the property of subtraction of segments in geometry.
Given:
 ABCD is a straight line.
 $AC = BD$.
We want to show that $AB = CD$.
Proof:
 Since ABCD is a straight line, the points A, B, C, and D lie on the same line in that order.
 Given that $AC = BD$.
 Now, consider the entire length of the line segment ABCD which is $AD$. We can express $AD$ in two different ways:
 $AD = AB + BC + CD$
 $AD = AC + CD = BD + AB$ (since $AC = BD$ by given and $AC$ incorporates $AB + BC$ while $BD$ incorporates $BC + CD$).
 By rearranging the above expression, we find that to maintain the equality, since the segments are parts of the same straight line and thus must sum up to the total length in any breakdown, we conclude $AB + BC + CD = AB + BC + CD$.
 This implies that both sides of the equation are equal, and thus, by the property of subtraction (subtracting equal parts from equal wholes), subtracting $BC$ from both sides, we obtain $AB = CD$.
Hence, it is proved that if $AC = BD$, then $AB = CD$ for a straight line ABCD.
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.)
Axiom 5, in the context of Euclid's axioms (not to be confused with the controversial fifth postulate about parallel lines), states that "The whole is greater than the part." This axiom is considered a 'universal truth' for several reasons that are fundamental to both mathematics and our understanding of quantities in the real world:

Intuitive Understanding: The axiom aligns with our intuitive and everyday experience. When we combine parts to make a whole, the complete entity invariably possesses a greater quantity, size, or value than any of its individual components.

Foundation for Measurement: This axiom underlies the concept of measurement and addition in mathematics. It ensures that when we add quantities together, the sum is larger than any of the individual quantities. This principle is critical for the coherence of arithmetic and algebra.

Logical Basis for Quantity Comparison: It provides a logical basis for comparing quantities, facilitating the understanding that adding to a quantity will result in a larger quantity. This is essential for developing further mathematical concepts such as inequalities, ratios, and proportions.

Universality across Disciplines: Its truth is not just confined to geometry or mathematics; it's observed universally across various fields, including physics, economics, and even biology. For example, in physics, when energies or forces combine, the total energy or force is greater than any individual contribution.

Abstract and Concrete Validity: The axiom holds true both in abstract mathematical systems and in concrete, realworld scenarios. This universality and versatility make it foundational to not just geometry but to the organization of knowledge itself.
Euclid's Axiom 5, by capturing a fundamental aspect of quantity and its aggregation, serves as a cornerstone for many mathematical arguments and processes, embodying a principle that is universally observable and inherently understandable.
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Ask Chatterbot AIExtra Questions  Introduction To Euclid’s Geometry  NCERT  Mathematics  Class 9
'Things which are equal to the same thing are equal to one another.' What kind of statement is this?
A. Axiom
B. Postulate
C. Theorem
D. Rule
The correct answer is A. Axiom.
The statement, "Things which are equal to the same thing are equal to one another," does not require a special proof because it is inherently obvious or selfevident. In Euclidean geometry, Euclid himself categorized such statements as axioms because they are accepted universally without proof.
Who is regarded as the father of Geometry?
A) Pythagoras
B) Euclid
C) Archimedes
D) Thales
Answer: B) Euclid
Explanation:
Euclid, a prominent Greek mathematician, is universally acknowledged as the "father of Geometry." He authored "Elements," a comprehensive compilation that encompasses various foundational geometric theories. These theories and principles laid out by Euclid continue to be fundamental in the field of mathematics globally.
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