Understanding Elementary Shapes

Introduction

All the shapes we see around us are formed using curves or lines. We can see corners, edges, planes, open curves and closed curves in our surroundings. We organise them into line segments, angles, triangles, polygons and circles. We find that they have different sizes and measures. Let us now try to develop tools to compare their sizes.

Measuring Line Segments

We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. To compare any two line segments, we find a relation between their lengths. This can be done in several ways.

(i) Comparison by observation:

 By just looking at them can you tell which one is longer? You can see that \(\overline{AB}\) is longer. But you cannot always be sure about your usual judgment. For example, look at the adjoining segments :

The difference in lengths between these two may not be obvious. This makes other ways of comparing necessary. In this adjacent figure, \(\overline{AB}\) and \(\overline{PQ}\) have the same lengths. This is not quite obvious. So, we need better methods of comparing line segments.

(ii) Comparison by Tracing

To compare AB and CD, we use a tracing paper, trace CD and place the traced segment on AB. Can you decide now which one among AB and CD is longer? The method depends upon the accuracy in tracing the line segment. Moreover, if you want to compare with another length, you have to trace another line segment. This is difficult and you cannot trace the lengths everytime you want to compare them.

(iii) Comparison using Ruler and a Divider

 Have you seen or can you recognise all the instruments in your instrument box? Among other things, you have a ruler and a divider.

Note how the ruler is marked along one of its edges. It is divided into 15 parts. Each of these 15 parts is of length 1cm.

Each centimetre is divided into 10subparts. Each subpart of the division of a cm is 1mm

How many millimetres make one centimetre? Since 1cm = 10 mm, how will we write 2 cm? 3mm? What do we mean by 7.7 cm? Place the zero mark of the ruler at A. Read the mark against B. This gives the length of AB. Suppose the length is 5.8 cm, we may write, Length AB = 5.8 cm or more simply as AB = 5.8 cm. There is room for errors even in this procedure. The thickness of the ruler may cause difficulties in reading off the marks on it.

Can we avoid this problem?

Is there a better way?

Let us use the divider to measure length. Open the divider. Place the end point of one of its arms at A and the end point of the second arm at B. Taking care that opening of the divider is not disturbed, lift the divider and place it on the ruler. Ensure that one end point is at the zero mark of the ruler. Now read the mark against the other end point

Understanding Elementary Shape

Right And Straight Angles

You have heard of directions in Geography. We know that China is to the north of India, Sri Lanka is to the south. We also know that Sun rises in the east and sets in the west. There are four main directions. They are North (N), South (S), East (E) and West (W). Do you know which direction is opposite to north? Which direction is opposite to west? Just recollect what you know already. We now use this knowledge to learn a few properties about angles. Stand facing north.

Do This

Turn clockwise to east. We say, you have turned through a right angle. Follow this by a ‘right-angle-turn’, clockwise.

You now face south.

 If you turn by a right angle in the anti-clockwise direction, which direction will you face? It is east again! (Why?)

 Study the following positions :

From facing north to facing south, you have turned by two right angles. Is not this the same as a single turn by two right angles?

The turn from north to east is by a right angle.

The turn from north to south is by two right angles; it is called a straight angle. (NS is a straight line!)

 Stand facing south.

Turn by a straight angle. Which direction do you face now? You face north! To turn from north to south, you took a straight angle turn, again to turn from south to north, you took another straight angle turn in the same direction. Thus, turning by two straight angles you reach your original position.

Think, discuss and write

By how many right angles should you turn in the same direction to reach your original position?

Turning by two straight angles (or four right angles) in the same direction makes a full turn. This one complete turn is called one revolution. The angle for one revolution is a complete angle.

We can see such revolutions on clock-faces. When the hand of a clock moves from one position to another, it turns through an angle.

Suppose the hand of a clock starts at 12 and goes round until it reaches at 12 again. Has it not made one revolution? So, how many right angles has it moved? Consider these examples :

Understanding Elementary Shapes

Acute Obtuse and Reflex Angles

We saw what we mean by a right angle and a straight angle. However, not all the angles we come across are one of these two kinds. The angle made by a ladder with the wall (or with the floor) is neither a right angle nor a straight angle.

Think, discuss and write

Are there angles smaller than a right angle? Are there angles greater than a right angle? Have you seen a carpenter’s square? It looks like the letter “L” of English alphabet. He uses it to check right angles. Let us also make a similar ‘tester’ for a right angle.

Other names

*An angle smaller than a right angle is called an acute angle.These are acute angles.

Do you see that each one of them is less than one-fourth of a revolution? Examine them with your RA tester.

* If an angle is larger than a right angle, but less than a straight angle, it is called an obtuse angle.These are obtuse angles.

Do you see that each one of them is greater than one-fourth of a revolution but less than half a revolution? Your RA tester may help to examine.

Identify the obtuse angles in the previous examples too.

*A reflex angle is larger than a straight angle. It looks like this. (See the angle mark) Were there any reflex angles in the shapes you made earlier? How would you check for them?

Understanding Elementary Shapes

Measuring Angles

The improvised ‘Right-angle tester’ we made is helpful to compare angles with a right angle. We were able to classify the angles as acute, obtuse or reflex. But this does not give a precise comparison. It cannot find which one among the two obtuse angles is greater. So in order to be more precise in comparison, we need to ‘measure’ the angles. We can do it with a ‘protractor’.

The measure of angle

We call our measure, ‘degree measure’. One complete revolution is divided into 360 equal parts. Each part is a degree. We write 360° to say ‘three hundred sixty degrees’.

The Protractor

You can find a readymade protractor in your ‘instrument box’. The curved edge is divided into 180 equal parts. Each part is equal to a ‘degree’. The markings start from 0° on the right side and ends with 180° on the left side, and viceversa.

Suppose you want to measure an angle ABC

1. Place the protractor so that the mid point (M in the figure) of its straight edge lies on the vertex B of the angle.

2. Adjust the protractor so that BC is along the straight-edge of the protractor.

3. There are two ‘scales’ on the protractor : read that scale which has the 0° mark coinciding with the straight-edge (i.e. with ray \(\overline{BC}\) ).

4. The mark shown by \(\overline{BA}\) on the curved edge gives the degree measure of the angle. We write m ∠ABC= 40°, or simply ∠ABC= 40°

Understanding Elementary Shapes

Perpendicular Lines

When two lines intersect and the angle between them is a right angle, then the lines are said to be perpendicular. If a line AB is perpendicular to CD, we write AB⊥ CD  . Think, discuss and write If AB⊥ CD, then should we say that CD⊥ AB  also?

Perpendiculars around us!

You can give plenty of examples from things around you for perpendicular lines (or line segments). The English alphabet T is one. Is there any other alphabet which illustrates perpendicularity?

 Consider the edges of a post card. Are the edges perpendicular?

Let \(\overline{AB}\) be a line segment. Mark its mid point as M. Let MN be a line perpendicular to \(\overline{AB}\) through M. Does MN divide AB into two equal parts? MN bisects \(\overline{AB}\) (that is, divides \(\overline{AB}\) into two equal parts) and is also perpendicular to \(\overline{AB}\).

So we say MN is the perpendicular bisector of \(\overline{AB}\). You will learn to construct it later.

Understanding Elementary Shapes

Classification of Triangles

 Do you remember a polygon with the least number of sides? That is a triangle. Let us see the different types of triangle we can get.

Take some more triangles and verify these. For this we will again have to measure all the sides and angles of the triangles. The triangles have been divided into categories and given special names. Let us see what they are

Naming triangles based on sides

A triangle having all three unequal sides is called a Scalene Triangle [(c), (e)].

A triangle having two equal sides is called an Isosceles Triangle [(b), (f)]

A triangle having three equal sides is called an Equilateral Triangle [(a), (d)]

Classify all the triangles whose sides you measured earlier, using these definitions

Naming triangles based on angles

If each angle is less than 90°, then the triangle is called an acute angled triangle.

If any one angle is a right angle then the triangle is called a right angled triangle.

 If any one angle is greater than 90°, then the triangle is called an obtuse angled triangle

Name the triangles whose angles were measured earlier according to these three categories. How many were right angled triangles?

Understanding Elementary Shapes

Quadrilaterals

 A quadrilateral, if you remember, is a polygon which has four sides.

(a) Both of you will have a pair of 30°– 60°– 90° set-squares. Place them as shown in the figure. Can you name the quadrilateral described? What is the measure of each of its angles?

This quadrilateral is a rectangle.

One more obvious property of the rectangle you can see is that opposite sides are of equal length.

What other properties can you find?

(b) If you use a pair of 45°– 45°–90° set-squares, you get another quadrilateral this time. It is a square.

Are you able to see that all the sides are of equal length? What can you say about the angles and the diagonals? Try to find a few more properties of the square.

 (c) If you place the pair of 30° – 60° – 90° set-squares in a different position, you get a parallelogram.

 Do you notice that the opposite sides are parallel? Are the opposite sides equal? Are the diagonals equal?

(d) If you use four 30° – 60° – 90° set-squares you get a rhombus.

(e) If you use several set-squares you can build a shape like the one given here. Here is a quadrilateral in which a pair of two opposite sides is parallel.

 It is a trapezium.

Here is an outline-summary of your possible findings. Complete it.

Understanding Elementary Shapes

Polygons

So far you studied polygons of 3 or 4 sides (known as triangles and quardrilaterals respectively). We now try to extend the idea of polygon to figures with more number of sides. We may classify polygons according to the number of their sides.

You can find many of these shapes in everyday life. Windows, doors, walls, almirahs, blackboards, notebooks are all usually rectanglular in shape. Floor tiles are rectangles. The sturdy nature of a triangle makes it the most useful shape in engineering constructions.

Look around and see where you can find all these shapes.