Introduction to Euclid’s Geometry – Class IX

**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 Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.**

Solution:

(i) False. Infinitely many lines can pass through a point in different directions.

(ii) False. Only a line can pass through two distinct points.

(iii) True. A terminated line can be produced indefinitely on both the sides.

(iv)True. If two circles are equal, then their radii are equal.

(v) True. From the axiom that if two lines are separately, equal to a third lines then they are equal to reach other.

**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**

Solution:

(i) parallel lines : Two or more lines which never intersect each other in a plane and perpendicular distance between them is always constant then they are said to be parallel lines.

Terms which is to be defined first are:

Plane: A plane is flat surface on which geometric figures are drawn.

Line: A line is a collection of points which can extend in both direction and has only length not breadth.

(ii) perpendicular lines : When two lines intersect at each other at right angle in a plane then they are said to be perpendicular to each other.

Terms which is to be defined first are:

Line: A line is a collection of points which can extend in both direction and has only length not breadth.

(iii) line segment : A line segment is a part of a line with two end points and cannot be extended further.

Terms which is to be defined first are:

Line: A line is a collection of points which can extend in both direction and has only length not breadth.

Point: A point is a dot drawn on a plane surface and is dimensionless.

(iv) radius of a circle : The fixed distance between the centre and the circumference of the circle is called the radius of the circle

Terms which is to be defined first are:

Circle: A **circle** is the locus of all points equidistant from a central point.

Centre:Centre is the point that is equally distant from every point on the circumference of a circle or sphere.

circumference of a circle: The **circumference of a circle** is the distance around it *or* it is the distance around the edge of a circle

(v) square: A square is a quadrilateral in which all the four sides are equal and each internal angle is a right angle.

Terms which is to be defined first are:

Right angle: an angle of 90°, as in a corner of a square, or formed by dividing a circle into quarters.

**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.**

Solution:

Undefined terms in the postulates:

The postulates are consistent. They do not contradict to one another. Both postulates do not follow from Euclid’s postulates.

Yes. Postulates are consistent when we deal with these two situation:

(i) point C lies in between and on the line segment joining A and B

(ii) Point C not lies on the line segment joining A and B

no, they do not follow from Euclid’s postulates. They follow the axioms.

- 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.

Solution:

Given AC = CB

AC + AC = CB + AC [*equals are added to equal]*

BC + AC coincides with AB

2AC = AB

AC = ^{1}/_{2}AB

**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:

Let s assume that we have two midpoints C and D i.e., C and D are midpoints of AB

AC = ^{1}/_{2}AB and AD = ^{1}/_{2}AB

⇒AC = AD

This is only possible if C = D or if C coincides with D .

Thus, a line segment has only one midpoint.

In Fig. if AC = BD, then prove that AB = CD.

Solution:

Given AC = BD

From the figure,

AC = AB + BC

BD = BC + CD

According to Euclid’s axiom, when equals are subtracted from equals , remainders are also equal.

Subtracting BC both sides,

AB + BC – BC = BC + CD – BC

AB = CD

- 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.)

Solution:

Axioms 5: The whole is always greater than the part.

Take an example of a pizza. When it is whole it will measures 1 kg but when we took out a part from it and measures its weight it will come out lower than the previous one. So, the fifth axiom of Euclid is true for all the universal things. Thus, it is considered as universal truth.

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