In the study of geometry, symmetry plays a pivotal role in understanding the properties and characteristics of various shapes. This section delves into the concepts of line symmetry and rotational symmetry, focusing on how these symmetries are identified in different shapes. The aim is to equip students with the skills to recognise and analyse the symmetry properties of two-dimensional (2D) and three-dimensional (3D) figures, enhancing their geometric reasoning and spatial awareness.

Line Symmetry

Line symmetry, also known as reflection symmetry, occurs when a shape can be divided into two identical halves that are mirror images of each other. The line dividing the shape is called the line of symmetry.

• Identifying Line Symmetry:
  • To determine if a shape has line symmetry, try to draw a line (or lines) through the shape so that one side of the line is a mirror image of the other.
  • A shape can have more than one line of symmetry. For example, an equilateral triangle has three lines of symmetry, each drawn from a vertex to the midpoint of the opposite side.

Example 1: Consider a rectangle. It has two lines of symmetry, one horizontal and one vertical, dividing it into equal halves.

Example 2: An isosceles triangle has one line of symmetry through the vertex perpendicular to the base.

Rotational Symmetry

Rotational symmetry exists when a shape can be rotated (less than a full turn) around a central point and still look the same as it did before the rotation.

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• Order of Rotational Symmetry:
  • The order of rotational symmetry is the number of times a shape fits onto itself in one complete turn. For example, a square has rotational symmetry of order 4 because it looks the same four times as we rotate it 360 degrees.
  • The formula to determine the angle of rotation is $\dfrac{360°}{\text{order of rotational symmetry}}$.

Example 3: A regular hexagon has rotational symmetry of order 6, meaning it matches its original position six times as we rotate it 360 degrees. The angle of rotation would be $\frac{360°}{6} = 60°$.

Applying Concepts to Shapes

Square

• Line Symmetry: A square has four lines of symmetry, each passing through the midpoint of opposite sides or diagonally across corners.

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• Rotational Symmetry: A square also possesses rotational symmetry of order 4. This means if you rotate a square by $90°, 180°, 270°$, or $360°$, it appears unchanged.

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Circle

• Line Symmetry: A circle has infinite lines of symmetry because any diameter or line through the centre divides it into mirror images.

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• Rotational Symmetry: Similarly, a circle has rotational symmetry of infinite order, as it can be rotated by any angle and still look the same.

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Worked Examples

Question 1

Identify the number of lines of symmetry and the order of rotational symmetry for an equilateral triangle.

Solution:

• An equilateral triangle has 3 lines of symmetry, each drawn from a vertex to the midpoint of the opposite side.
• It has a rotational symmetry of order 3 since it can be rotated three times around its centre (at angles of $120°, 240°$, and $360°$) and still match its original position.

Question 2

A regular pentagon is placed in front of you. Determine its lines of symmetry and the order of rotational symmetry.

Solution:

• A regular pentagon has 5 lines of symmetry, each connecting a vertex to the midpoint of the opposite side.
• The order of rotational symmetry is 5, indicating that it can be rotated $72°, 144°, 216°, 288°$, and $360°$ to coincide with its original shape.

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Key Points to Remember

• Line Symmetry: A shape has line symmetry if it can be divided into two identical halves that are mirror images of each other.
• Rotational Symmetry: A shape exhibits rotational symmetry if it can be rotated around a central point by a certain angle and still appear the same.
• The order of rotational symmetry represents how many times a shape fits onto itself during a 360-degree rotation.