# Introduction To Euclid’s Geometry Chapter 5 Class 9 Maths NCERT PDF

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### Chapter 5: Introduction to Euclid’s Geometry

#### 5.1 Introduction

The word ‘geometry’ comes from the Greek words ‘geo’, meaning the ‘earth’, and ‘meter’, meaning ‘to measure’.

Geometry appears to have originated from the need for measuring land. This branch of mathematics was studied in various forms in every ancient civilization, be it in Egypt, Babylonia, China, India, Greece, the Incas, etc.

The people of these civilizations faced several practical problems which required the development of geometry in various ways.

For example, whenever the river Nile overflowed, it wiped out the boundaries between the adjoining fields of different landowners.

After such flooding, these boundaries had to be redrawn. For this purpose, the Egyptians developed a number of geometric techniques and rules for calculating simple areas and also for doing simple constructions.

Their knowledge of geometry was also used by them for computing volumes of granaries, and for constructing canals and pyramids.

They also knew the correct formula to find the volume of a truncated pyramid (see Fig. 5.1). You know that a pyramid is a solid figure, the base of which is a triangle, or square, or some other polygon and its side faces are triangles converging to a point at the top.

In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc. show that the Indus Valley Civilisation (about 3000 BCE) made extensive use of geometry.

It was a highly organized society. The cities were highly developed and very well planned. For example, the roads were parallel to each other and there was an underground drainage system. The houses had many rooms of different types.

This shows that the town dwellers were skilled in mensuration and practical arithmetic. The bricks used for construction were kiln fired and the ratio of length: breadth: thickness, of the bricks, was found to be 4: 2: 1.

In ancient India, the Sulbasutras (800 BCE to 500 BCE) were the manuals of geometrical constructions. The geometry of the Vedic period originated with the construction of altars (or vedis) and fireplaces for performing Vedic rites.

The location of the sacred fires had to be in accordance with the clearly laid down instructions about their shapes and areas if they were to be effective instruments.

Square and circular altars were used for household rituals, while altars whose shapes were combinations of rectangles, triangles, and trapeziums were required for public worship.

The Sri Yantra (given in the Atharvaveda) consists of nine interwoven isosceles triangles.

These triangles are arranged in such a way that they produce 43 subsidiary triangles. Though accurate geometric methods were used for the construction of altars, the principles behind them were not discussed.

### NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry

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 that pass through two distinct points.

(iii) A terminated line can be produced indefinitely on both 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

There can be an infinite number of lines that can be drawn through a single point. Hence, the statement mentioned is False

(ii) False

Through two distinct points, there can be only one line that can be drawn. Hence, the statement mentioned is False

(iii) True

A line that is terminated can be indefinitely produced on both sides as a line can be extended on both its sides infinitely. Hence, the statement mentioned is True.

(iv) True

The radii of two circles are equal when the two circles are equal. The circumference and the center of both the circles coincide; and thus, the radius of the two circles should be equal. Hence, the statement mentioned is True.

(v) True

According to Euclid’s 1st axiom- “Things which are equal to the same thing are also equal to one another”. Hence, the statement mentioned is True.

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

(v) square

Solution:

Yes, there are other terms that need to be defined first, they are:

Plane: Flat surfaces in which geometric figures can be drawn are known are planes. A plane surface is a surface that lies evenly with straight lines on it.

Point: A dimensionless dot that is drawn on a plane surface is known as a point. A point is that which has no part.

Line: A collection of points that has only length and no breadth is known as a line. And it can be extended in both directions. A line is a breadth-less length.

(i) Parallel lines – Parallel lines are those lines that never intersect each other and are always at a constant distance perpendicular to each other. Parallel lines can be two or more lines.

(ii) Perpendicular lines – Perpendicular lines are those lines that intersect each other in a plane at right angles then the lines are said to be perpendicular to each other.

(iii) Line Segment – When a line cannot be extended any further because of its two endpoints then the line is known as a line segment. A line segment has 2 endpoints.

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

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