Now, as we have already mentioned, a shape can have more than one line of symmetry. To illustrate this, let’s take four common shapes—a square, an equilateral triangle, a rectangle and a circle.

• • A Rectangle

    • Parallel sides are equal.

    • Adjacent sides are not equal.

    • Diagonals are equal. 

    • All angles are at 90°.

For a rectangle, the perpendicular bisectors of its sides act as lines of symmetry. If you cut a rectangle along these two lines, you will get two halves that mirror each other. No other line of symmetry is possible. Hence, a rectangle has only two lines of symmetry.

 

• • An Equilateral Triangle

    • Three sides, each of equal length.

    • All angles are at 60°.

   For an equilateral triangle, the three angle bisectors become the lines of symmetry.

• • A Square

    • All sides are equal.

    • Diagonals are equal.

    • All angles are at 90°.

    • Angles between diagonals are not at 90°.

   In addition to the perpendicular bisector of sides, you also have the diagonals for a square. Hence, a square has four       lines of symmetry.

• • A Circle

    • One centre.

    • Fixed radius.

    • Infinite diameters.

    • All points are equidistant from the centre.

   Every diameter divides a circle into two identical halves. Hence, every diameter acts as a line of symmetry for a circle.       Since a circle has infinite diameters, infinite lines of symmetry are possible for a circle.

Let us get to the questions now.

Problem 1: Find the number of lines of balance in the following shapes:

i. A Hexagon

   Solution: See the following figure:

   

The three lines of symmetry are possible for this figure.

       ii. A Swastika

There is no line of symmetry in a swastika.

Line symmetry and mirror reflection are very closely related.

For any mirror reflection, the object and its image are symmetrical. There is no change in the lengths and angles of the object. Only one thing changes—the lateral orientation.

The image formed is laterally inverted, which means the left becomes the right and vice-versa. Because of this, the image can completely overlap with the object. Some of its applications are as follows.

Paper decoration (Fun Exercise)

Fold a thin rectangular coloured paper several times and create intricate patterns by cutting the paper. When you unfold the paper, you will find various repeating patterns of cut-outs. Try to identify the lines of symmetry in these patterns.

Kaleidoscope

A Kaleidoscope is a device that uses inclined mirrors to form a V-shape. It produces images with various lines of symmetry. The angles between the mirrors can be adjusted and determine the number of symmetrical lines formed.

Rangoli Patterns

Rangoli patterns are trendy in India, especially on Diwali or Holi. Rangoli patterns also make extensive use of symmetry to achieve attractive shapes.

Let us do some questions now:

Problem 1: Find the number of lines of symmetry in these shapes.

 

 

Solution: There are two lines of symmetry, as shown.

Based on whether you slide, rotate or produce a translation motion in the object, we can classify symmetry into four major types.

These four types are:

1. Translation Symmetry: This kind of symmetry has an axis that moves an object or a geometrical shape. While their                  position keeps changing, the identical objects keep repeating. It is like driving an object along a straight-line path, also            called translation. For example, consider a circle. If we keep replicating that circle along the X-axis direction linearly, its            shape or size would not change.

2. Rotational Symmetry: As the name suggests, rotational symmetry employs rotation. We keep rotating a given object            until it regains its identical position. If we take a square and rotate it by 90 degrees, we get the initial shapes. If you turn a        square by 180, 270 or 360 degrees even, the symmetry is maintained. Hence, a square has four rotational symmetries.

3. Reflexive Symmetry: For this symmetry to occur, one half of the object must be the mirror reflection of the other. For            example, take any diameter of a circle. If you place a mirror on the diameter, the image of one part of the circle obtained        will be the same as the other.

4. Glide Symmetry: This is a special kind of symmetry. It is a combination of two different types: translation and reflection          symmetry. For example, a circle.

Now that we have discussed most of the syllabus, here are some more familiar shapes and their lines of symmetry for you.

 

Problem 1: Can you draw a triangle that has:

a) Exactly one line of symmetry?

-   An isosceles triangle.

b) Exactly two lines of symmetry?

-   None possible.

c) Exactly three lines of symmetry?

-   An equilateral triangle.

 d) No lines of symmetry?

-   A scalene Triangle.

Problem 2: In the letters of the English alphabet (A-Z), list the ones that have the following:

a) Horizontal lines of symmetry

-   B, C, D, E, H, I, K, O, X

b) Vertical lines of symmetry

-   A, H, I, M, O, T, U, V, W, X, Y

c) No lines of Symmetry

-    F, G, J, L, N, P, Q, R, S

Problem 3: Suppose you are standing 100 cm in front of the mirror. How far behind the mirror will your reflection be? If you start moving closer to the mirror, how will your reflection move?

-   The reflection will be 100 cm behind the mirror. And it will start moving towards the mirror at the same speed.

Symmetry is one of the most fundamental geometry topics for high school students, and nature also vouches for it. From the reflection of trees in clear water to rotational symmetry in flowers, nature keeps showing it everywhere. In mathematics, symmetry comes handy in understanding shapes better. However, even alphabets of English, Hindi, or any other language have symmetry. As such, students must have an in-depth understanding of it.