Basic Geometrical Ideas

Introduction

Geometry has a long and rich history. The term ‘Geometry’ is the English equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’ means Measurement. According to historians, the geometrical ideas shaped up in ancient times, probably due to the need in art, architecture and measurement. These include occasions when the boundaries of cultivated lands had to be marked without giving room for complaints. Construction of magnificent palaces, temples, lakes, dams and cities, art and architecture propped up these ideas. Even today geometrical ideas are reflected in all forms of art, measurements, architecture, engineering, cloth designing etc.

You observe and use different objects like boxes, tables, books, the tiffin box you carry to your school for lunch, the ball with which you play and so on. All such objects have different shapes. The ruler which you use, the pencil with which you write are straight. The pictures of a bangle, the one rupee coin or a ball appear round. Here, you will learn some interesting facts that will help you know more about the shapes around you.

Basic Geometrical Ideas

Points

By a sharp tip of the pencil, mark a dot on the paper. Sharper the tip, thinner will be the dot. This almost invisible tiny dot will give you an idea of a point.

A point determines a location. These are some models for a point : If you mark three points on a paper, you would be required to distinguish them. For this they are denoted by a single capital letter like A,B,C.

These points will be read as point A, point B and point C. Of course, the dots have to be invisibly thin.

Basic Geometrical Ideas

A Line Segment

Fold a piece of paper and unfold it. Do you see a fold? This gives the idea of a line segment. It has two end points A and B.

 Take a thin thread. Hold its two ends and stretch it without a slack. It represents a line segment. The ends held by hands are the end points of the line segment

The following are some models for a line segment :

Try to find more examples for line segments from your surroundings. Mark any two points A and B on a sheet of paper. Try to connect A to B by all possible routes. (Fig 4.1) What is the shortest route from A to B?

This shortest join of point A to B (including A and B) shown here is a line segment. It is denoted by \(\overline{AB}\) or \(\overline{BA}\). The points A and B are called the end points of the segment.

Basic Geometrical Ideas

A Line

 Imagine that the line segment from A to B (i.e. \(\overline{AB}\)) is extended beyond A in one direction and beyond B in the other direction without any end (see figure). You now get a model for a line

Do you think you can draw a complete picture of a line? No. (Why?) A line through two points A and B is written as\[\(\overline{AB}\)]. It extends indefinitely in both directions. So it contains a countless number of points. (Think about this). Two points are enough to fix a line. We say ‘two points determine a line’. The adjacent diagram (Fig 4.3) is that of a line PQ written as\(\overline{PQ}\). Sometimes a line is denoted by a letter like l, m.

Basic Geometrical Ideas

Intersecting Lines

Look at the diagram (Fig 4.4).

Two lines \(l_1\) and \(l_2\) are shown. Both the lines pass through point P. We say \(l_1\) and \(l_2\)intersect at P. If two lines have one common point, they are called intersecting lines. The following are some models of a pair of intersecting lines (Fig 4.5) :

Try to find out some more models for a pair of intersecting lines.

Basic Geometrical Ideas

Parallel Lines

 Let us look at this table (Fig 4.6). The top ABCD is flat. Are you able to see some points and line segments? Are there intersecting line segments?

Yes, AB and BC intersect at the point B. Which line segments intersect at A? at C? at D?

Do the lines \(\overline{AD}\)  and \(\overline{CD}\) intersect?

You find that on the table’s surface there are line segment which will not meet, however far they are extended. \(\overline{AD}\) and \(\overline{BC}\) form one such pair. Can you identify one more such pair of lines (which do not meet) on the top of the table?

 Lines like these which do not meet are said to be parallel; and are called parallel lines.

Think, discuss and write

Where else do you see parallel lines? Try to find ten examples. If two lines \(\overline{AB}\) and \(\overline{CD}\) s ruu are parallel, we write \(\overline{AB}\) || \(\overline{CD}\). If two lines \(l_1\) and \(l_2\) are parallel, we write  \(l_1\) || \(l_2\).

Can you identify parrallel lines in the following figures?

Ray

The following are some models for a ray : A ray is a portion of a line. It starts at one point (called starting point or initial point) and goes endlessly in a direction. Look at the diagram (Fig 4.7) of ray shown here. Two points are shown on the ray. They are (a) A, the starting point (b) P, a point on the path of the ray. We denote it by \(\overline{AP}\)

Basic Geometrical Ideas

Curves

Have you ever taken a piece of paper and just doodled? The pictures that are results of your doodling are called curves

You can draw some of these drawings without lifting the pencil from the paper and without the use of a ruler. These are all curves (Fig 4.10). ‘Curve’ in everyday usage means “not straight”. In Mathematics, a curve can be straight like the one shown in fig 4.10 (iv). Observe that the curves (iii) and (vii) in Fig 4.10 cross themselves, whereas the curves (i), (ii), (v) and (vi) in Fig 4.10 do not. If a curve does not cross itself, then it is called a simple curve.

Draw five more simple curves and five curves that are not simple.

 Consider these now (Fig 4.11). What is the difference between these two? The first i.e. Fig 4.11 (i) is an open curve and the second i.e. Fig 4.11(ii) is a closed curve. Can you identify some closed and open curves from the figures Fig 4.10 (i), (ii), (v), (vi)? Draw five curves each that are open and closed.

Position in a figure

A court line in a tennis court divides it into three parts : inside the line, on the line and outside the line. You cannot enter inside without crossing the line. A compound wall separates your house from the road. You talk about ‘inside’ the compound, ‘on’ the boundary of the compound and ‘outside’ the compound. In a closed curve, thus, there are three parts. (i) interior (‘inside’) of the curve (ii) boundary (‘on’) of the curve and (iii) exterior (‘outside’) of the curve.

In the figure 4.12, A is in the interior, C is in the exterior and B is on the curve. The interior of a curve together with its boundary is called its “region”.

Basic Geometrical Ideas

Polygons

 Look at these figures 4.13 (i), (ii), (iii), (iv), (v) and (vi).

What can you say? Are they closed? How does each one of them differ from the other? (i), (ii), (iii), (iv) and (vi) are special because they are made up entirely of line segments. Out of these (i), (ii), (iii) and (iv) are also simple closed curves. They are called polygons. So, a figure is a polygon if it is a simple closed figure made up entirely of line segments. Draw ten differently shaped polygons.

Do This

Try to form a polygon with

1. Five matchsticks.

2. Four matchsticks.

3. Three matchsticks.

4. Two matchsticks.

In which case was it not possible? Why?

Sides, vertices and diagonals

Examine the figure given here (Fig 4.14).

Give justification to call it a polygon. The line segments forming a polygon are called its sides. What are the sides of polygon ABCDE? (Note how the corners are named in order.)

Sides are \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{DE}\) and \(\overline{EA}\) .

The meeting point of a pair of sides is called its vertex

Sides \(\overline{AE}\) and \(\overline{ED}\) meet at E, so E is a vertex of the polygon ABCDE. Points B and C are its other vertices. Can you name the sides that meet at these points? Can you name the other vertices of the above polygon ABCDE?

Any two sides with a common end point are called the adjacent sides of the polygon.

Are the sides \(\overline{AB}\) and \(\overline{BC}\) adjacent? How about \(\overline{AE}\) and \(\overline{DC}\)? The end points of the same side of a polygon are called the adjacent vertices. Vertices E and D are adjacent, whereas vertices A and D are not adjacent vertices. Do you see why?

Consider the pairs of vertices which are not adjacent. The joins of these vertices are called the diagonals of the polygon.

In the figure 4.15, \(\overline{AC}\), \(\overline{AD}\), \(\overline{BD}\), \(\overline{BE}\) and \(\overline{CE}\)are diagonals. Is \(\overline{BC}\) a diagonal, Why or why not?

If you try to join adjacent vertices, will the result be a diagonal? Name all the sides, adjacent sides, adjacent vertices of the figure ABCDE (Fig 4.15). Draw a polygon ABCDEFGH and name all the sides, adjacent sides and vertices as well as the diagonals of the polygon.

Basic Geometrical Ideas

Angles

Angles are made when corners are formed.

Here is a picture (Fig 4.16) where the top of a box is like a hinged lid. The edges AD of the box and AP of the door can be imagined as two rays \(\overline{AB}\) and \(\overline{AP}\) . These two rays have a common initial point A. The two rays here together are said to form an angle. An angle is made up of two rays starting from a common initial point. The two rays forming the angle are called the arms or sides of the angle. The common initial point is the vertex of the angle.

This is an angle formed by rays \(\overline{OP}\) and \(\overline{OQ}\) (Fig 4.17). To show this we use a small curve at the vertex. (see Fig 4.17). O is the vertex. What are the sides? Are they not \(\overline{OP}\) and \(\overline{OQ}\) ? How can we name this angle? We can simply say that it is an angle at O. To be more specific we identify some two points, one on each side and the vertex to name the angle. Angle POQ is thus a better way of naming the angle. We denote this by ∠POQ

Think, discuss and write

Look at the diagram (Fig 4.18).What is the name of the angle? Shall we say ∠P ? But then which one do we mean? By ∠P what do we mean? Is naming an angle by vertex helpful here? Why not? By ∠P we may mean ∠APB or ∠CPB or even ∠APC! We need more information

Do This

Take any angle, say ∠ABC

Shade that portion of the paper bordering \(\overline{BA}\) and where \(\overline{BC}\) lies.

Now shade in a different colour the portion of the paper bordering \(\overline{BC}\)and where \(\overline{BA}\) lies.

The portion common to both shadings is called the interior of ∠ABC(Fig 4.19). (Note that the interior is not a restricted area; it extends indefinitely since the two sides extend indefinitely).

In this diagram (Fig 4.20), X is in the interior of the angle, Z is not in the interior but in the exterior of the angle; and S is on the ∠PQR . Thus, the angle also has three parts associated with it.