Introduction To Euclid’s Geometry - Class 9 - Mathematics
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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
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Sign up nowConsider 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.
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Sign up nowIf 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 mid-point of line segment $A B$. Prove that every line segment has one and only one mid-point.
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Sign up nowIf ABCD is a straight line, if $A C=B D$, then prove that $A B=C D$.
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Sign up nowWhy 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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