1. Symmetry

By studying this lesson you will be able to

• Identify plane figures with bilateral symmetry,

• Draw the axes of symmetry of a bilaterally symmetric figure, and

• Create bilaterally symmetric figures on square ruled paper.

1.1 Bilateral Symmetry

A few figures having two parts which coincide with each other when folded along a certain line are shown below.

Many of the objects in the environment have the property that they can be divided into two equal parts. Most creations too have this property which helps preserve their beauty. Let us learn more about plane figures and laminas with a plane figure as the boundary, that have this property.

In figure 1, there is only one line that divides the figure into two equal parts which coincide. However, each of the figures 2, 3 and 4, has more than one line that divides the figure into two parts which coincide.

If a plane figure can be folded along a straight line so that you get two parts which coincide, then that plane figure is defined as a bilaterally symmetric plane figure. The line of folding is defined as an axis of symmetry of the figure.

In a bilaterally symmetric figure, the two parts on either side of an axis of symmetry are of the same shape and of the same area.

A line through a plane figure which divides it into two parts of the same shape and of the same area which do not coincide with each other is not an axis of symmetry of the figure.

1.2 Drawing axes of symmetry

(1) From the following, choose the bilaterally symmetric figures with a correctly drawn axis of symmetry and write down the corresponding letters.

(2) (i) Cut out laminas of the following shapes using a tissue paper. Draw all the axes of symmetry of each of them.

(ii) Paste all the figures having axes of symmetry in your exercise book.

(3) (i) Cut laminas of the following shapes using paper. Draw all the axes of symmetry of each of them.

A - Rectangular shape

B - Triangular shape with two sides of equal length

(ii) Write the number of axes of symmetry in each of the above figures.

(iii) Create another symmetric figure by joining two figures of the shapes given in A and B above and paste it in your exercise book.

(4) Write the statements below in your exercise book. Mark a  in front of the correct sentences and a  in front of the incorrect ones.

(i) In a bilaterally symmetric figure, the two parts on either side of an axis of symmetry are equal in shape and in area.

(ii) There are bilaterally symmetric figures having more than one axis of symmetry.

(iii) The number of axes of symmetry in a circular lamina is greater than the number of axes of symmetry in a square.

(iv) The maximum number of axes of symmetry in a bilaterally symmetric figure is one.

(v) If a bilaterally symmetric figure which has at least two axes of symmetry is cut along one axis and divided into two equal parts, then each of these parts too will be bilaterally symmetric.

1.3 Creating plane figures having bilateral symmetry

At the end of the above activity you obtain a bilaterally symmetric figure. Its axis of symmetry is the initial line along which you folded the paper.

Assignment

💠Create various bilaterally symmetric plane figures by cutting out folded paper as well as by placing drops of paint on folded paper as done in the previous activities.

💠Prepare an attractive wall decoration using the symmetric figures that you created.

1.4 Drawing bilaterally symmetric plane figures

Let us consider the symmetric plane figure given below which has been drawn on a square ruled paper.

The axis of symmetry of this figure is the vertical line indicated by the dotted line. The points at which the straight line segments of a rectilinear plane figure meet are defined as the vertices of the plane figure. Usually the vertices are named using capital letters of the English alphabet.

The vertices A, B, C and D are on the right side of the axis of symmetry of the figure. Let us consider where the points A' , B' , C' and D' are located on the left side of the axis of symmetry.

The point A' is located at a distance from the axis of symmetry which is equal to the distance from A to the axis of symmetry, on a horizontal line which passes through A. A' is defined as the vertex corresponding to A.

Similarly, B' , C' and D' are defined as the vertices corresponding to B, C and D respectively.

Let us consider how a bilaterally symmetric figure is drawn on a square ruled paper (or grid) by identifying corresponding vertices.

Now you have obtained a bilaterally symmetric rectilinear plane figure with the dotted line as its axis of symmetry and the marked points as vertices.

Let us consider how symmetric figures can be drawn by using the above properties.

(3) Copy each of the figures given below in your exercise book. Complete the figures so that you obtain a bilaterally symmetric figure in each case.

(4) Trace each of the figures given below on a tissue paper and copy them in your exercise book.

Now turn the tissue paper on the dotted line. Draw the other half of each of the figures to obtain bilaterally symmetric figures.

(5) 

(i) Draw three bilaterally symmetric figures on a square ruled paper such that each figure has only one axis of symmetry.

(ii) Draw the axis of symmetry of each of the above figures.

(6) 

(i) Draw two bilaterally symmetric figures on a square ruled paper such that each figure has only 2 axes of symmetry.

(ii) Draw the axes of symmetry of each figure.

Summary

• If a plane figure is divided into two equal parts which coincide with each other when folded along a particular line, then that figure is defined as a bilaterally symmetric figure.

• The line of folding described above is an axis of symmetry of the figure.