Chapter 3 Introduction to Euclid’s Geometry

**NCERT Solutions for Class 9 Maths Chapter 3 Introduction to Euclid’s Geometry Ex 3.1**

**Question 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 figure, if AB – PQ and PQ = XY, then AB = XY.**

**Solution:**

**(i)** False. In a single point, infinite number of lines can pass through it.

**(ii)** False. For two distinct points only one straight line is passing.

**(iii)** True.

**(iv)** True. [∵ Radii of congruent (equal) circles are always equal]

**(v)** True. AB = PQ …..(i)

PQ = XY

⇒ XY = PQ …(ii)

From Eqs. (i) and (ii), we get AB = XY

**Question 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**

**Solution:**

**(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.

**(ii)** Perpendicular lines Two lines in a plane are said to be perpendicular, if they intersect each other at one right angle.

In figure, P and Q are said to be perpendicular lines because they ; intersect each other at 90° and we write Q ⊥ P.

Here, the term one right angle is undefined.

**(iii)** Line segment The definite length between two points is called the line segment.

In figure, the definite length between A and B is line represented by AB¯¯¯¯¯¯¯¯.

Here, the term definite length is undefined.

**(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 adjoining figure OA is the radius.

Here, the term, point and centre is undefined.

**(v)** Square A square is a rectangle having same length and breadth.

Here, the terms length, breadth and rectangle are undefined.

**Question 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**.

**Solution:**

There are so many undefined words which should be knowledge. They are consistent because they deal with two different situations that is

(i) if two points A and B are given, then there exists a third point C which is in between A and B.

(ii) if two points A and B are given, then we can take a point C which don’t lie on the line passes through the point A and B.

These postulates don’t follow Euclid’s postulates. However, they follow axiom Euclid’s postulate 1 stated as through two distinct points, there is a unique line that passes through them.

**Question 4.**

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

**Solution:**

Line AB is drawn by joining A and B using between a point C is taken. AC is taken and put on line AB.

i.e., AC covers the line segment AB in two overlaps.

∴ AB = AC + AC

AB = 2 AC

⇒ AC = 12 AB

**Question 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:**

Here, C is the mid-point of line segment AB, such that AC = BC

Let there are two mid-points C and C’ of AB.

⇒ AC = 12 AB

and AC’ = 12 AB

⇒ AC = AC’

which is only possible when C and C’ coincide.

⇒ Point C’ lies on C.

Hence, every line segment has one and only one mid-point.

**Question 6.**

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

**Solution:**

We have

AC = BC

⇒ AC – BC = BD – BC (∵ Equals are subtracted from equals)

⇒ AB = CD

**Question 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.)**

**Solution:**

According to axiom 5, we have The whole is greater than a part, which is a universal truth.

Let a line segment PQ = 8 cm. Consider a point R in its interior, such that PR = 5 cm

Clearly, PR is a part of the line segment PQ and ft lies in its interior.

⇒ PR is smaller than PQ.

Hence, the whole is greater than its part.

**NCERT Solutions for Class 9 Maths Chapter 3 Introduction to Euclid’s Geometry Ex 3.2**

**Question 1.**

**How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?**

**Solution:**

Two distinct intersecting lines cannot be parallel to the same line.

**Question 2.**

**Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.**

**Solution:**

Yes.

According to Euclid’s fifth postulate when line x falls on line y and z such that ∠1+ ∠2< 180°. Then, line y and line z on producing further will meet in the side of ∠1 arid ∠2 which is less than 180°.

We find that the lines which are not according to Euclid’s fifth postulate. i.e., ∠1 + ∠2 = 180°, do not intersect.