NCERT solution for class 9 maths introduction to euclidean geometry ( Chapter 5)

Solution for Exercise 5.1

1.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) In figures, if AB=PQ and PQ=XY, then AB=XY.
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Answer :

i)False, infinite number of lines can pass through a single point.
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ii)False, only one straight line can pass through two distinct points. Say points are P and Q.
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iii)True, a terminated line can be produced indefinitely on both the sides.
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iv)True, since, radii of congruent (equal) circles are always equal i.e., if you superimpose the region bounded by one circle on the other, then they coincide.

v)True, if AB = PQ and PQ = XY i.e. XY = PQ
Then, AB = XY.

2.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 tines
(iii) Line segment
(iv) Radius of a circle
(v) Square
Answer :

i)Parallel lines: Two lines in a plane are said to be parallel, if they have no point in common.
In figure, x and y are said to be parallel because they have no point in common and we write, x||y . Here, the term point is undefined.
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ii)Perpendicular lines: Two lines in a plane are said to be perpendicular, if they intersect each other at a right angle.
In figure, P and Q are said to be perpendicular lines because they intersect each other at \(90^\circ\).
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iii)Line segment: The definite length between two points is called the line segment. ln figure, the definite length between A and B is line segment and represented by AB.
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iv)Radius of a circle: The distance from the centre to a point on the circle is called the radius of the circle. ln the adjoining figure OA is the radius.
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v)Square: A square can be defined as a rectangle having same length and breadth.

3. 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 atleast 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.
Answer :

There are so many undefined words which should be knowledge.
They are consistent because they deal with two different situations i.e.,
(i) If two points A and B are given, then there exists a third point C which is lying in between A and B on the line.
(ii) If two points A and B are given, then we can take a point C which don’t lie on the line passing through the point A and B.
These postulates do not follow Euclid’s postulate.
However, they follow axiom Euclid’s postulate 1 stated as "through two distinct points, there is a unique line that passes through them."

4: 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.
Answer :

Line AB is drawn by joining A and B and a point C is taken between them such that AC covers the line segment AB in two overlaps.
AB = AC + BC
But, AC = BC,
Therefore, AB = 2AC
i.e., AC = 1/2 AB.
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5.In question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.
Answer :

Here, C is the mid-point of line segment AB, such that AC = BC.
Let us assume that there are two mid-points C and C' of same line AB.
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Then as already proved, AC = 1/2 AB and also, AC' = 1/2 AB
Therefore, we get, AC = AC' ,which is only possible when C and C’ Coincide.
Hence, it is proved that every line segment has one and only one mid-point.

6.In figure, if AC = BD, then prove that AB = CD.
image
Answer :

Given that,
AC = BD ....(i)
AC = AB + BC as point B lies between A and C ....ii)
BD = BC + CD as point C lies between B and D ....iii)
From equations (i), (ii) and (iii), we get,
AB + BC = BC + CD
Subtracting equals we get,
AB = CD.

7.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.)
Answer :

According to axiom 5, we have 'The whole is greater than a part', which is a universal truth.
Let a line segment PQ = 8cm. Consider a point R in its interior, such that PR = 5cm.
Clearly, PR is a part of the line segment PQ and R lies in its interior.
Hence, PR is smaller than PQ.
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Hence, the whole is greater than its part is proved.

Solution for Exercise 5.2

1.How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
Answer :

Any valid formulation by student can be considered.
But still, it can be rewritten as two distinct intersecting lines cannot be parallel to the same line or parallel line cannot be intersecting.

2.Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
Answer :

Yes, according to Euclid’s fifth postulate when line x falls on straight line y and z such that sum of the interior angles on one side of line x is two right angles.
i.e., \(\angle{1}\) + \(\angle{2}\) = \(180^\circ\).
Then, line y and line z on producing further will meet in the side of \(\angle{1}\) and \(\angle{2}\) which is less than \(180^\circ\).
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We find that the lines which are not according to Euclid’s fifth postulate. i.e., \(\angle{1}\) + \(\angle{2}\) = \(180^\circ\), do not intersect.
So, the lines y and z never meet and are, therefore, parallel.