# Class 7 – Symmetry

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Symmetry
• Line and rotational symmetry
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## Study Material

 Introduction: Similar patterns add beauty to an object. Patterns are nothing but similar shapes/ objects. So, in this synops is you will learn how symmetry of a shape/ object is taken.   Symmetry: We have already learnt in the lower classes about linear symmetry, some different shapes, figures and objects having one or more lines of symmetry, reflection and rotational symmetry. A plane figure which is identical on both sides of a line is said to be symmetrical about that line and that line is called axis of symmetry or line of symmetry or mirror line. In order to check, whether a given figure is symmetrical about a line or not, fold the figure about that line. If part of the figure which lies on one side of the line exactly coincides with the part of the figure on the other side of the line, then the figure is symmetrical about that line; otherwise not symmetrical or asymmetrical. Study the following figures in which line of symmetry are marked with dotted lines: Lines of symmetry for regular polygons: Let us discuss lines of symmetry for various regular polygons. We know that a polygon is a closed figure made of several line segments. The polygon made up of the least number of line segments is called a triangle. A polygon is said to be regular if all its sides are of equal length and all its angles are of equal measure. Thus, an equilateral triangle is a regular polygon of three sides. An equilateral triangle is regular because each of its sides has same length and each of its angles measures 600. A square is also regular because all its sides are of equal length and each of its angles is a right angle. Its diagonals are seen to be perpendicular bisector of one another.

If a pentagon is regular, then its sides should have equal length. We will later on learn that the measure of each of its angles is 1080.

A regular hexagon has all its sides equal and each of its angles measures 1200.

The regular polygons are symmetrical figures and hence their lines of symmetry are quite interesting. Each regular polygon has as many lines of symmerty as it has sides. We say, they have multiple lines of symmetry.

The concept of line symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it, is the mirror image of other half. We see the mirror reflection in the following figures.

Example: Find the line (s) of symmetry of the following figures with punched holes shown by small circles:

Solution: The line (s) of symmetry corresponding to given punched holes are shown by dotted line:

Example: Given the line(s) of symmetry, find the placement of the other hole (s):

Solution: With respect to the given line (s) of symmetry, the other hole(s) are marked in the given figures as under.

Rotational symmetry: Some figures are symmetric about their rotation. If a point of the figure is fixed such that the figure can be rotated about that point, then the point is called centre of rotation.

A figure is said to be rotationally symmetric if after certain degree of rotation, it looks the same as it is its starting position.

i) A figure can have more than one rotational symmentry.

ii) If a figure has ‘n’ different rotations leading to rotational symmetry then we say that the figure has rotational symmetry of order ‘ n’.

For example: In the adjoining figure is shown an equilateral triangle.

Let this figure be rotated through  ( clockwise or anticlockwise ) about its centre, then in the resulting position, it will again look same as it did in its starting position.

Therefore, an equilateral triangle possesses rotational symmetry.

Rotation of the figure through and also give us the same result. Thus, in total, we have three distinct rotations of the equilateral triangle in which it looks the same as it was in its original position. Thus, we say that equilateral triangle has a rotational symmetry of order 3.

i) If  is the smallest angle through which a figure can be rotated and still looks the same, then it has a rotational symmetry of order (n) .

As in the above illustration, order of rotational symmetry .

ii) For a figure to have rotational symmetry angle must be less than or equal to 1800.

Rotational symmetry of some figures:

i) A line segment has rotational symmetry of order 2.

ii) An angle, a scalene triangle, an isosceles triangle, an isosceles trapezium, a kite, an arrowhead, a semi-circle, each has no rotational symmetry.

iii) A parallelogram, a rectangle and a rhombus each have rotational symmetry of order 2.

iv) An equilateral triangle has rotational symmetry of order 3. The centre of rotation is the point of intersection of the bisectors of the angles (Incentre).

v) A square has rotational symmetry of order 4.

vi) A regular pentagon has rotational symmetry of order 5.

vii) A regular hexagon has rotational symmetry of order 6.

viii) A circle has rotational symmetry of infinite order.

ix) The following letters of English alphabet, each has rotational symmetry of order 2.

Example: Give the order of the rotational symmetry of the given figures about the marked point X:

Solution: (i) To find the order of rotation in respect of X. Let us mark the end of vertical rectangle as A. We find that figure required for rotations through 900 each about the point (x) to get back to the original position. Thus, this figure has a rotational symmetry of order 4.

(ii) To find the order of rotation in respect of ‘A’

Clearly, this figure requires three rotations, each through an angle of , about the marked point (x) to come back to its original position. Therefore, it has rotational symmetry of order 3.

Example: Give the order of rotational symmetry for each figure:

Solution: Let us mark a point A on each figure and also indicate the angle through which the figure is to be rotated as under:

∴ You see that the order of rotational symmetry for the above figures is as follows.

i) n = 2       ii) n = 2      iii) n = 3 and                iv) n = 4

A ‘n’ sided regular polygon have ‘n’ lines of symmetry as well as order of rotational symmetry ‘n’.

Example:

4 lines of symmetry

Order of rotational symmetry = 4.