Line Symmetry and Rotational Symmetry: Definitions, Facts, Examples

We often come across the word SYMMETRY in our day to day life; as we will say, there is a kind of symmetry in butterfly wings, symmetry in architectural buildings, and so on. Do you know what symmetry is?? When a figure is folded into two halves such that both the halves are identical, we say that the figure is symmetrical, and thus, this phenomenon is called symmetry. The line which divides the figure into two identical halves is called the line of symmetry. This article will explain in detail the two main types of symmetries, i.e. line symmetry and rotational symmetry.

Line Symmetry and Rotational Symmetry

Line Symmetry: Fold a rectangular piece of paper as shown in figure (a).

Then cut a piece of any design from the folded side of the paper as shown in figure (b).

Learn the Concepts on Rotational Symmetry

Now, a design emerges when we unfold the paper, as shown in figure (c).

The design pattern is identical on both sides of the folding line of the paper, as shown in figure (c) by the dotted line \(AB\). Thus, \(AB\) is the line of symmetry here. A figure that is identical on both sides of a line is said to be symmetrical about that line, and the line about which the figure is symmetrical is called the line of symmetry or the axis of symmetry.

To find whether or not a given figure is symmetrical about a line in it, fold the figure about that line. If the figure on one side of the line coincides with the figure on the other side, the figure is symmetrical about that line.

Rotational Symmetry: Many objects rotate about a fixed point. For example, a windmill, wheels of a vehicle, hands of a clock, blades of a fan, etc. When an object rotates, its shape and size do not change. Thus, a shape is said to have a rotational symmetry when it looks the same even after a rotation. Rotation can be clockwise or anti-clockwise. Hence, rotational symmetry is the property of a shape or object that looks the same after some rotation by partial turn.

Thus, a figure is said to have a rotational symmetry if it fits onto itself more than once during a complete rotation.

Here, we are rotating a square. Whatever rotation is given, the square is looking the same. Hence the square has rotational symmetry.

Order of Rotational Symmetry

The number of distinct orientations in which the shape looks the same as the original is called its order of rotational symmetry. A complete turn or full turn means rotation through \(360^\circ \). In a complete turn, there are mainly \(4\) rotational positions.

1. Rotation through \(90^\circ \).
2. Rotation through \(180^\circ \).
3. Rotation through \(270^\circ \).
4. Rotation through \(360^\circ \).

The above rotation are, respectively, called as 

1. Quarter turn
2. Half turn
3. Three-fourth turn, and 
4. Full turn

Figures Having Line Symmetry and Rotational Symmetry

By now, we have understood that some shapes have line symmetry, and some shapes have rotational symmetry. Some shapes have both, i.e. line symmetry as well as rotational symmetry. We will now discuss the symmetry of some objects, which exhibits both types of symmetry.

Square: When a square \(A B C D\) is rotated about the point \(O\) through \(90^{\circ}, 180^{\circ}, 270^{\circ}\) and \(360^{\circ}\) will fit exactly each time onto itself. So it has rotational symmetry of order \(4\). Also, a square has \(4\) lines of symmetry, namely, the diagonals and the lines of joining the mid-points of opposite sides.

Rectangle: As shown below, a rectangle \(A B C D\) fits exactly each time onto itself when rotated through \(180^{\circ}\) and \(360^{\circ}\). So, it has rotational symmetry of order \(2\). Also, it has \(2\) lines of symmetry.

Equilateral Triangle: An equilateral triangle \(A B C\) fits exactly each time onto itself when rotated through \(120^{\circ}, 240^{\circ}\) and \(360^{\circ}\) about the centroid \(O\). So, it has rotational symmetry of order \(3\). Also, it has \(3\) lines of symmetry along the bisectors of interior angles of the triangle.

Lines of Symmetry and No Rotational Symmetry

Sometimes rotational symmetry can be hard to spot. So, we can either turn the book or the paper where the shape is drawn on or trace it onto the tracing paper and turn it on top of the given shape. Then, we will be able to find the shape match. 

A kite only looks the same once during a complete turn, so it does not have rotational symmetry.

An isosceles triangle has a line of symmetry but does not have rotational symmetry. Look at the below-given figure. It has \(3\) lines of symmetry. Also, it has rotational symmetry of order \(3\) when rotated through \(120^{\circ}, 240^{\circ}\) and \(360^{\circ}\).

Learn the Concepts of Line Symmetry

Solved Examples – Line Symmetry and Rotational Symmetry

Q.1. Show the position of a line segment \(\overline{A B}\) after rotation about a point \(O\) through \(90^{\circ}\) clockwise.

Ans: Let the given line segment \(\overline{A B}\) is rotated clockwise about the point \(O\). \(O\) is the centre of the rotation. Join \(O A\) and \(O B\) with dotted lines.
Now, let us draw \(\angle A O A^{\prime}=90^{\circ}\) and cut off at \(O A^{\prime}=O A\). Again draw \(\angle B O B^{\prime}=90^{\circ}\) and cut off \(O B^{\prime}=O B\). Join \(A\)’ and \(B\)’. Thus, \(A^{\prime} B^{\prime}\) is the image of \(A B\) after a rotation of \(90^{\circ}\) clockwise about \(O\).

Q.2. Draw the lines of symmetry for the given figure and find the number of lines of symmetry.

Ans: For the given figure, we can draw a horizontal and vertical line of symmetry.

Thus, the given figure will have \(2\) lines of symmetry.

Q.3. Rotate an equilateral \(\triangle A B C\) about its centre and find the order of rotational symmetry for it.
Ans: An equilateral triangle acquires its original position when rotated through \(120^{\circ}, 240^{\circ}\) and \(360^{\circ}\).

The original shape is obtained
a) After a rotation of \(120^\circ \)
b) After a rotation of \(240^\circ \)
c) After a rotation of \(360^\circ \)
One complete rotation is obtained after a rotation of \(360^{\circ}\). An equilateral triangle looks like the original after every rotation through \(120^{\circ}\). This angle is called the angle of rotation. Here, the angle of rotation \(=120^{\circ}\). Therefore, the order of rotational symmetry \(\frac{360^{\circ}}{120^{\circ}}=3\). Thus, the order of the rotational symmetry \(=3\).

Q.4. Complete the table.

Ans: a) Equilateral Triangle: An equilateral triangle has \(3\) sides.
It has \(3\) lines of symmetry.

b) Square: A square has \(4\) sides.

A square has \(4\) lines of symmetry.

c)  Regular pentagon: A regular pentagon has \(5\) sides

A regular pentagon has \(5\) lines of symmetry.

Regular hexagon: A regular hexagon has \(6\) sides.

A regular hexagon has \(6\) lines of symmetry.

Regular polygons have the same order of rotational symmetry as the number of sides. Hence, the complete table is as follows;

Q.5. Let \(O\) be the fixed point or the centre of the rotation. Show the position of \(^{*}\) after every rotation through \(90^{\circ}\) clockwise.

Ans: a) Position of \(*\) after a clockwise rotation through \(90^{\circ}\) is

b) Position of \(*\) after a clockwise rotation through \(180^\circ \) is

c) Position of \(*\) after a clockwise rotation through \(270^{\circ}\) is

e) Position of \(*\) coincides with itself after a clockwise rotation through \(360^{\circ}\).

Summary

In this article, we learned in detail about line symmetry and rotational symmetry. We also learnt to find the order of rotational symmetry. In addition to this, we learnt about the shapes that have line symmetry and rotational symmetry. Lastly, we learned to solve some line and rotational symmetry examples to strengthen our grip on the concept.

Frequently Asked Questions (FAQs)

Q.1. What is line symmetry and rotational symmetry?
Ans: A figure that is identical on both sides of a line is said to be a symmetrical figure. When an object is rotated and looks the same as the original figure, the figure has rotational symmetry.

Q.2. What is the difference between line symmetry and line of symmetry?
Ans: The line which divides the figure into two identical halves is called the line of symmetry. If the shape of an object is the same when a line is drawn through the middle of it, then the shape has line symmetry.

Q.3. What is rotational symmetry in math?
Ans: A shape is said to have rotational symmetry when it looks the same even after a rotation. Rotation can be clockwise or anti-clockwise. When an object or a shape rotates, its size and shape do not change. Instead, it undergoes a turning movement about a specific point known as the centre of the rotation. This phenomenon is known as rotational symmetry.

Q.4. What are the three types of symmetry in math?
Ans: The \(3\) types of symmetry in math are
a) Line Symmetry
b) Rotational symmetry
c) Translational symmetry

Q.5. What is a rotational symmetry example?
Ans: The example of rotational symmetry is a windmill, wheels of a vehicle, hands of a clock, blades of a fan, etc.

Now you are provided with all the necessary information on the concepts of line symmetry and rotational symmetry and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.