NCERT Class 9 Solutions: Introduction To Euclid's Geometry (Chapter 5) Exercise 5.1 – Part 3

Euclid discribed the following axioms and postulates. An axiom is a statement which is strongly self-evident. A "postulate," on the other hand, is simply postulated, i.e. it is assumbed to be true:

Euclid’s Axioms

  • Things which are equal to the same thing are also equal to one another.

  • If equals are added to equals, the whole are equal.

  • If equals be subtracted from equals, the remainders are equal.

  • Things which coincide with one another are equal to one another.

  • The whole is greater than the part.

Euclid’s Postulates

  • A straight line segment can be drawn joining any two points.

    A line segment AB

    A Line Segment AB

    A line segment AB

  • Any straight line segment can be extended indefinitely in a straight line.

    Line AB extends infinitely in both directions

    Line AB

    Line AB extends infinitely in both directions

  • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

    Circle wih center O and radius R

    Circle Wih Center O and Radius R

    Circle wih center O and radius R

  • All right angles are congruent.

    Understanding right angle as quater of a cirlcle

    Right Angle

    Understanding right angle as quater of a cirlcle

  • If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles ( 180° ), then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. In the below figure α+β<180° , thus the lines intersect on that side.

    Lines intersect because angle alpha + angle beta is less than 180 degree

    Parallel Postulate

    Lines intersect because angle alpha + angle beta is less than 180 degree

Q-3 Consider the two ‘postulates’ given below:

  1. Given any two distinct points A and B, there exists a third point C which is in between A and B.

  2. 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.

Solution:

Yes the postulates contain the terms point and line which are not defined by the postulates themselves but are evident. These ‘postulates’ do not follow from Euclid’s postulates.

  1. That the given two points A and B, there is a point C lying on the line in between them. This postulates follows from Euclid’s postulates that a line can be drawn between two points and we know that there are infinite number of points on a line.

    Note that this says that there are infinite many points on a line. Or in other words an ideal point is infinitely small.

  2. That given A and B, we can make a C not lying on the line through A and B. This follows from the Euclid’s postulate that, “given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center”. Now assume that A and B are diameter of these circle. Than any other point on the circle (except A and B) will not be on line segment AB.

    All points on the circle except A and B are not on the line segment AB.

    Circle With Diameter AB

    All points on the circle except A and B are not on the line segment AB.

Q-4 If point C lies between two points A and B such that AC = BC, then prove that AC = ½ AB. Explain by drawing the figure.

Solution:

According to Euclid’s axioms “Things which coincide with one another are equal to one another.”

Line segment AB with C in mid-point

Line Segment AB With C in Mid-Point

Line segment AB with C in mid-point

Now since C lies between points A and B AC and CB will coincide and hence AC = BC. Also we know that AC + BC = AB. Since BC = AC we can write, AC + AC = AB, that is 2AC = AB or AC= 12 AB

Q-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.

Solution:

Line segment with supposedly two midpoints. Proof using contradiction

Line Segment With Supposedly Two Midpoints

Line segment with supposedly two midpoints. Proof using contradiction

  • We know that AC=BC

  • Let D be another mid-point of AB. Then, AD=DB ……………equation (ii)

  • Subtracting the above two equations, ACAD=BCDB , we can do this because of Euclid’s axioms “If equals be subtracted from equals, the remainders are equal.”

  • Since ACAD=DC and CBDB=DC , therefore we get DC = -DC. DC+DC=2DC=0DC=0 . In other words C and D coincide.

Q-6 In the following figure, if AC=BD , then prove that AB=CD

A line with AC = BD

Line With AC = BD

A line with AC = BD

Solution:

  • Given, AC=BD(1) and AC=AB+BC(2) point B is within the line segment AC.

  • Also BD=BC+CD(3) since point C lies between B and D

  • Now, substituting equation (2) and (3) in equation (1), we get

AB+BC=BC+CD

AB+BCBC=CD (subtracting both sides by BC)

AB=CD

Hence, AB=CD.

Q-7 Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the 5th postulate)

Solution:

The 5th axiom is “The whole is always greater than the part.” The whole is greater than the part, could be interpreted as a definition of “greater than.” To say one magnitude B is a part of another A could be taken as saying that A is the sum of B and C for some third magnitude C, the remainder. Symbolically, A > B means that there is some C such that A = B + C. Here B and C are “parts” of A and A is the “whole”. The order or “greater than and less than” are the basic universal truths of the nature.

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