Objective:

This topic gives an overview of; 

• Introduction of symmetry
• Lines of Symmetry for Regular Polygons
• Irregular polygon

Introduction

Symmetry is an important geometrical concept, commonly exhibited in nature and is used almost in every field of activity. Artists, professionals, designers of clothing or jewellery, car manufacturers, architects and many others make use of the idea of symmetry. The beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs — everywhere you find symmetrical designs.

You have already had a ‘feel’ of line symmetry in your previous class. A figure has a line symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide. You might like to recall these ideas. There are some activities to help you.

Let us now strengthen our ideas on symmetry further. Study the following figures in which the lines of symmetry are marked with dotted lines.

Lines of Symmetry for Regular Polygons

A polygon is a closed figure made of several line segments. The polygon made up of the least number of line segments is the triangle. (There can be a polygon that we can draw with still fewer line segments).

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 60°.

A square is also regular because all its sides are of equal length and each of its angles is a right angle (i.e., 90°). Its diagonals are seen to be perpendicular bisectors of one another .

If a pentagon is regular, naturally, its sides should have equal length. You will, later on, learn that the measure of each of its angles is 108° .

A regular hexagon has all its sides equal and each of its angles measures 120°.

The regular polygons are symmetrical figures and hence their lines of symmetry are quite interestingEach regular polygon has as many lines of symmetry 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 the other half . A mirror line, thus, helps to visualise a line of symmetry .

While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation .

Irregular polygon:

Most irregular polygons do not have line symmetry. However, some of them do. Look at the rectangle and the isosceles triangle. A rectangle has two lines of symmetry, and an isosceles triangle has one line of symmetry.

Some letters have line symmetry. The letters A, B, C, D, E, I, K, M, T, U, V, W and Y have one line of symmetry.

The letter H overlaps perfectly both vertically and horizontally. So it has two lines of symmetry. Similarly, the letter X has two lines of symmetry. The letters F, G, J, L, N, P, Q, R, S and Z have no line of symmetry.

Summary:

• A figure has line symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide.
• Regular polygons have equal sides and equal angles. They have multiple (i.e., more than one) lines of symmetry.
• Each regular polygon has as many lines of symmetry as it has sides.
• Mirror reflection leads to symmetry, under which the left-right orientation have to be taken care of.
• Most irregular polygons do not have line symmetry.
• The letters A, B, C, D, E, I, K, M, T, U, V, W and Y have one line of symmetry.The letters F, G, J, L, N, P, Q, R, S and Z have no line of symmetry.