CBSE Notes Class 7 Maths Chapter 12 Symmetry

If we fold a picture in half and both the halves-left half and right half-match exactly then we can say that the picture is symmetrical. Eg.– India Gate and Red Fort in Delhi, Taj Mahal in Agra, are all symmetrical monuments. Symmetry has different types, such as reflection symmetry, line symmetry, rotational symmetry etc.

1.0Transformation

Transformation geometry shows how shape changes position and size according to certain rules. Some of the most common mathematical transformations are reflection (flips), rotations (turns), translation (sliding without turning), and enlargement or reductions (making larger or smaller).

2.0Line Symmetry

A shape is symmetric if it can be folded so that both halves match exactly, with the fold line being the axis of symmetry, also known as the mirror line. The two halves of a symmetrical figure are identical in shape and size.

Linear symmetry in some common geometrical shapes

• Lines: A line has infinite symmetry lines, all perpendicular to it, aligning with its image.

• Segments: A segment has two symmetry lines—the perpendicular bisector and the segment itself.

• Angles: An angle has one symmetry line, its angular bisector.

Shapes and their symmetry lines

• Isosceles triangle: One symmetry line, the angular bisector of the vertex and the perpendicular bisector of the base. 

• Equilateral triangle: Three symmetry lines, each an angular bisector. 

• Isosceles trapezium: One symmetry line, the perpendicular bisector of the parallel sides.

• Kites and arrowheads: One symmetry line along a diagonal.

• Rhombus: Two symmetry lines, both diagonals.

• Rectangle: Two symmetry lines, the perpendicular bisectors of opposite sides.

• Square: Four symmetry lines—two diagonals and perpendicular bisectors of opposite sides.

• Circle: Infinite symmetry lines through the center (diameters).

Symmetry of Letters of the Alphabet

The following letters of the English alphabet are symmetrical about the dotted line (or lines).

The following letters have no line of symmetry.

3.0Reflection Symmetry (Mirror Symmetry)

A mirror creates an image of the same size and shape but reversed, called a reflection. Reflection symmetry acts like a mirror, where each point on the image is the same perpendicular distance from the axis of reflection as its corresponding point on the object. The image of a point P reflected over a line ℓ (the mirror line) is called P′ if ℓ is the perpendicular bisector of PP′. This transformation, P → P′, is a reflection in ℓ. 

Key points:

• The line ℓ is the mirror line.
• Points on ℓ are their own images.
• Reflections preserve length and angle measures.
• The image of a reflected figure is congruent to the original.
• Perpendicular and parallel lines remain the same after reflection.

A glide reflection combines translation followed by reflection in a parallel line.

1. Reflection in x-axis

A point P (a, b) has its image P´(a, –b), when reflected in x-axis (y = 0).

2. Reflection in y-axis

A point P (a, b) has its image P´(–a, b), when reflected in y-axis (x = 0).

4.0Rotational Symmetry

Rotation involves turning an object around a fixed point, called the centre of rotation, by a certain angle in either a clockwise or anticlockwise direction. The angle of rotation measures how far the object turns. A full rotation is 360°, a half-turn is 180°, and a quarter-turn is 90°. The object's shape and size remain unchanged during rotation.

Clockwise rotation follows the movement of a clock's hands, while the opposite direction is anticlockwise.

Centre of rotation and angle of rotation

Rotation maintains an object's shape and size while turning it around a fixed point called the centre of rotation. The angle of rotation indicates how far the object turns: a full turn is 360°, a half turn is 180°, and a quarter turn is 90°.

1. Rotation through 90°

Plot a point on a graph-paper and rotate the graph paper through 90° about the origin O. 

1. When rotated through 90° anticlockwise, point P(a, b) takes the position P’(–b, a). 
2. When rotated through 90° clockwise, point P(a, b) takes the position P’’(b, –a).

2. Rotation through 180°

Plot a point on a graph-paper and rotate the graph-paper through 180° about the origin O. When rotated through 180°, the point P(a, b) takes the position P´(–a, –b). Both directions of rotation (clockwise/anticlockwise), produce the same result.

5.0Line symmetry and Rotational symmetry

We now discuss the symmetry of some geometrical shapes which have both line of symmetry as well as rotational symmetry.

A Square

Square ABCD has rotational symmetry of order 4 as it aligns with itself when rotated 90°, 180°, 270°, and 360° about point O. It also has 4 lines of symmetry: the diagonals and the lines connecting the midpoints of opposite sides. 

A Rectangle

Rectangle ABCD has rotational symmetry of order 2, fitting onto itself when rotated 180° and 360° about point O. It also has 2 lines of symmetry.

An Equilateral Triangle

Equilateral triangle ΔABC has rotational symmetry of order 3, fitting onto itself when rotated 120°, 240°, and 360° about its centroid O. It also has 3 lines of symmetry along the angle bisectors.