ICSE Class 8 Maths Geometry Chapter 08 Symmetry Reflection and Rotation

Symmetry, Reflection and Rotation

Linear Symmetry

A body is said to be symmetrical or exhibit symmetry when different parts of it matches exactly in shape and size. When this symmetry occurs across a line drawn through the middle of a figure, or when a straight line divides a plane figure into two parts that are identical in size and shape, we say that the figure has linear symmetry.

In the figure, the line AB divides the mask into two identical halves. Such a figure is called a linear symmetric figure and the line AB is called the line (or the axis) of symmetry or the mirror line. A figure may have one or more lines of symmetry, as shown below.

This pentagon has 1 line of symmetry

A rectangle has 2 lines of symmetry

An equilateral triangle has 3 lines of symmetry

A square has 4 lines of symmetry

Teacher's Note

Symmetry is everywhere in nature - from butterfly wings to snowflakes. Understanding symmetry helps us recognize patterns and appreciate the balance found in everyday objects.

Some Symmetrical Figures and Their Lines of Symmetry

Teacher's Note

Regular shapes like squares and circles have multiple symmetry lines, while irregular shapes may have none. This concept helps in art, design, and understanding the structure of molecules in chemistry.

Example - Draw an axis of symmetry for the following (if possible).

(i) A shape resembling the letter H

(ii) A shape resembling the letter T

(iii) A shape resembling the letter I

(iv) A trapezoid

(v) A parallelogram

(vi) A circle with a smaller circle inside

Solution

(i) The H shape has 2 lines of symmetry - one vertical through the middle and one horizontal through the middle.

(ii) The T shape has 1 line of symmetry - a vertical line through the middle.

(iii) The I shape has 2 lines of symmetry - one vertical through the middle and one horizontal through the middle.

(iv) The trapezoid has 1 line of symmetry - a vertical line through the middle of the parallel sides.

(v) A parallelogram has no axis of symmetry.

(vi) The circle with a smaller circle inside has infinite lines of symmetry - any line passing through both centers.

Rotational Symmetry

A plane figure is said to have rotational symmetry if it remains the same after being rotated about a central point through an angle less than 360 degrees. Needless to say that any figure will coincide with itself when rotated through 360 degrees.

If the adjoining figure is rotated about its central point P through 360 degrees, it will coincide with its outline twice once after a rotation of 180 degrees and them after another rotation of 180 degrees. We therefore say that it has rotational symmetry of order 2. Thus, the order of rotational symmetry of a figure is the highest number of times that the figure coincides with its outline when rotated through 360 degrees about a fixed point called its central point.

Teacher's Note

Rotational symmetry appears in wheels, pinwheels, and clock hands. Understanding rotation helps in physics when studying circular motion and in design when creating logos that look the same from multiple angles.

Alternative method

If x degrees is the smallest angle of rotation that makes a figure coincide with its outline then the order of rotational symmetry of the figure equals \[\frac{360}{x}\]. Thus, the order of rotational symmetry of the adjoining figure is 4.

Hence, the adjoining figure has rotational symmetry of order 4.

Example - Find the central point and the order of rotational symmetry of

(i) a square

(ii) an equilateral triangle

(iii) H

(iv) a rhombus

(v) S

(vi) N

Solution

(i) The central point O is the point of intersection of the diagonals of the square. The order of rotational symmetry equals 4 (angles of rotation equals 90 degrees, 180 degrees, 270 degrees, 360 degrees).

(ii) The central point is the centroid of the equilateral triangle. The order of rotational symmetry equals 3 (angles of rotation equals 120 degrees, 240 degrees, 360 degrees).

(iii) The central point is the point O. The order of rotational symmetry equals 2 (angles of rotation equals 180 degrees, 360 degrees).

(iv) The central point is the point of intersection of the diagonals. The order of rotational symmetry equals 2 (angles of rotation equals 180 degrees, 360 degrees).

(v) The central point is the point O as shown. The order of rotational symmetry equals 2 (angles of rotation equals 180 degrees, 360 degrees).

(vi) The central point is the point O as shown. The order of rotational symmetry equals 2 (angles of rotation equals 180 degrees, 360 degrees).

This is a preview of the first 3 pages. To get the complete book, click below.