Exercise 5.1

Q1

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 the below figure, if AB = PQ and PQ = XY, then AB = XY.

Sol

Solution:

(i) FalseTrue

There can be infinitefew number of lines that can be drawn through a single point.

(ii) FalseTrue

Through two distinct points, there can be only onetwo line that can be drawn.

(iii) TrueFalse

A line that is terminated can be indefinitely produced on bothone side(s) as a line can be extended on both its sides infinitely.

(iv) TrueFalse

The radii of two circles are when the two circles are equal. The circumference and the centre of both the circles coincideoverlap; and thus, the radius of the two circles should be .

(v) TrueFalse

According to Euclid’s 1st axiom- “Things which are equal to the same thing are also to one another”.

Q2

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 lines (iii) line segment

(iv) radius of a circle (v) square

Sol

Solution:

Yes, there are other terms which need to be defined first. They are as follows:

Plane: Flat surfaces in which geometric figures can be drawn are known as .

A plane surface is a surface which lies evenly with straight lines on it.

Point: A dimensionless1D dot which is drawn on a plane surface is known as point.

A point is that which has no part.

Line: A collection of points that has only length and noone breadth is known as a line. It can be extended in bothone directions.

A line is breadth-lessheight-less length.

(i) Parallel lines: Parallel lines are those lines which never intersectintersect at a point each other and are always at a constant perpendicularhorizontal distance between each other.

Parallel lines can be two (or) more lines.

(ii) Perpendicular lines: Perpendicular lines are those lines which intersect each other in a plane at rightacute angles.

(iii) Line segment: When a line cannot be extended any further because of its twomultiple end points, then the line is known as a line segment. A line segment has end points.

(iv) Radius of circle: A radius of a circle is the line from any point on the of the circle to the of the circle.

(v) Square: A quadrilateral in which all the four sides are said to be , and each of its internal angles is a angle, is called square.

Q3

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 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.

Sol

Solution:

YesNo, these postulates containdo not contain undefined terms.

Undefined terms in the postulates are as follows:

(a) There are many points that lie in a plane. But, in the postulates given here, the position of the point C is not givengiven, as of whether it lies on the line segment joining AB or not.

(b) On top of that, there is novery little information about whether the points are in same plane or not.

So, they don’t followfollow from Euclid’s postulates. They followdon’t follow the axioms.

Q4

4. If a point C lies between two points A and B such that AC = BC, then prove that AC = 12AB. Explain by drawing the figure.

Sol

Solution:

Given: AC = BC

Adding AC both sides: AC + AC = BC + AC

AC = BC + AC

We know that, BC + AC = ABBC

∴ 2 AC = AB (If equals are added to equals, the wholes are equalunequal.)

⇒ AC = 122 AB.

Hence, proved.

Q5

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.

Sol

Solution:

Let AB be the line segment. Assume that points P and Q are the two different mid points of AB. Therefore,

AP = PBQB and AQ = QBPB.

Also, PB + AP = ABBP (as it coincidesjoins with line segment AB)

Similarly, QB + AQ = ABBQ

Adding AP to the LHS and RHS of the equation: AP = PB

We get: AP + AP = PB + AP (If equals are added to equals, the wholes are equal.)

⇒ AP = AB — (i)

Similarly, AQ = AB — (ii)

From (i) and (ii), since RHS are same, we equate the LHS.

2 AP = 2 AQ (Things which are to the same thing are equal to one another.)

⇒ AP = AQQP (Things which are double of the same things are equal to one another.)

Thus, we conclude that P and Q are the samedifferent points.

This contradicts our assumption that P and Q are two different mid points of AB.

Thus, it is proved that every line segment has one and only one mid-point.

Q6

6. In the below figure, if AC = BD, then prove that AB = CD.

Sol

Solution:

Given: AC = BD

From the given figure, we get:

AC = AB + BC

BD = BC + CD

⇒ AB + BC = BC + CD as AC = BDAD

We know that, according to Euclid’s axiom, when equals are subtracted from equals, remainders are also equalnot equal.

Subtracting BC from the LHS and RHS of the equation AB + BC = BC + CD, we get: AB+BC−BC = BC+CD−BC

Thus, AB =

Hence, proved.

Q7

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.)

Sol

Solution:

Axiom 5: The whole is always greaterlessersometimes greater than the part.

For Example: A cake. When it is whole or complete, assume that it measures 2 pounds but when a part from it is taken out and measured, its weight will be smaller than the previous measurement. So, the fifth axiom of Euclid is true for allsome of the materials in the universe.

Hence, Axiom 5, in the list of Euclid’s axioms, is considered a ‘universal truth’.