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Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

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Q1. 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 that pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both sides.
(iv) If two circles are equal, then their radii are equal.
(v) In the following figures, if AB = PQ and PQ = XY, then AB = XY.

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

Ans: (i) False

[If we mark a point O on the surface of a paper and using pencil and ruler, we can draw an indefinite number of straight lines passing through O.]

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

(ii) False

[∵ In the following figure, there are many straight lines passing through ‘P’. There are many lines, passing through ‘Q’. But there is one and only one line which is passing through ‘P’ as well as ‘Q’.]

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

(iii) True

[∵ Postulate 2 says that “A terminated line can be produced indefinitely.”]

(iv) True

[∵ Superimposing the region of one circle on the other, we find them coinciding. So, their centres and boundaries coincide. Thus, their radii will coincide.]

(v) True

[∵ According to Euclid’s axiom, things that are equal to the same thing are equal to one another.]

Q2. 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
Ans: 
Yes, we need to have an idea about the terms, point, line, ray, angle, plane, circle and quadrilateral, etc. before defining the required terms.
Point: A small dot made by a sharp pencil on the surface of paper gives an idea about a point.
It has no dimensions. It has only a position.
Line: A line is an idea that it should be straight and that it should extend indefinitely in both directions. It has no endpoints and has no definite length.

Note: In geometry, a line means “The line in its totality and not a portion of it. Whereas a physical example of a perfect line is not possible. A line extends indefinitely in both directions, so we cannot draw or show it wholly on paper. That is why we mark arrowheads on both ends, indicating that it extends indefinitely in both directions.

Ray: A part of a line that has only one endpoint and extends indefinitely in one direction. A ray has no definite length.
Angle: Two rays having a common endpoint form an angle.
Plane: A plane is a surface such that every point of the line joining any two points on it, lies on it.
Circle: A circle is the set of all those points in a given plane that are equidistant from a fixed point in the same plane. The fixed point is called the centre of the circle.
Quadrilateral: A closed figure made of four line segments is called a quadrilateral.
Definitions of the required terms are given below: 

(i) Parallel Lines: Two lines ‘n’ and ‘m’ in a plane are said to be parallel if they have no common point and we write them as n || m.

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

Note: The distance between two parallel lines always remains the same.

(ii) Perpendicular Lines: Two lines ‘p’ and ‘q’ lying in the same plane are said to be perpendicular if they form a right angle and we write them as p ⊥ q.

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

(iii) Line Segment: A line segment is a part of the line having a definite length. It has two end-points.
In the figure, a line segment is shown having endpoints ‘A’ and ‘B’. It is written as  Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9or Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

(iv) Radius of a circle: The distance from the centre to a point on the circle is called the radius of the circle. In the figure, P is the centre and Q is a point on the circle, then PQ is the radius.

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

(v) Square: A quadrilateral in which all the four angles are right angles and all the four sides are equal is called a square. In the figure, PQRS is a square.

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9


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 Euclid’s postulates? Explain.
Ans: Yes, these postulates contain undefined terms such as Point and Line.
Also, these postulates are consistent because they deal with two different situations as (i) says that given two points ‘A’ and ‘B’. There is a point ‘C’ lying on the line in between them. Whereas (ii) says that, given ‘A’ and ‘B’, you can take ‘C’ not lying on the line through ‘A’ and ‘B’.

No, these postulates do not follow Euclid’s postulates, however, they follow from the axiom: “Given two distinct points, there is a unique line that passes through them.”


Q4. If a point C lies between two points A and B such that AC = BC, then prove that AC =  (1/2)AB. Explain by drawing the figure.

Ans: Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

∵ AC = BC      [given]
∴ AC + AC = BC + AC        [If equals added to equals then wholes are equal]
or 2AC = AB      [∵ AC + BC = AB]
or AC = (1/2)AB

Q5. In question 4, point C is called a mid-point of the segment AB. Prove that every line segment has one and only one mid-point.
Ans: Let the given line AB is having two midpoints ‘C’ and ‘D’.

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9     ...(1)
and Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9  ...(2)
From (1) and (2), we have
Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9
or AD – AC = 0
or CD = 0
∴ C and D coincide.
Thus, every line segment has one and only one mid-point.

Q6. In the figure, given below, if AC = BD, then prove that AB = CD.
Ans: We have:

Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9

AC = BD                      [Given]     ..(1)
Since, the point B lies between A and C,
∴ AC = AB + BC        ...(2)
Similarly, the point C lies between B and D,
∴ BD = CD + BC        ...(3)
From (1), (2) and (3)
AB + BC = CD + BC
[If equals are subtracted from equals, the remainders are equal]
⇒ 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.)
Ans: In the given list of Euclid’s axioms, we have:
“The whole is greater than the part”
This statement is true for all things and in all parts of the Universe.
So, it is a ‘Universal truth’.

The document Ex 5.1 NCERT Solutions - Introduction to Euclid’s Geometry Notes | Study Mathematics (Maths) Class 9 - Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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