International Mathematics (0607) | Symmetry

International Mathematics 0607: Geometry – Symmetry

Hello Future Mathematicians!

Welcome to the chapter on Symmetry! This topic is all about balance, patterns, and how shapes map exactly onto themselves. Geometry isn't just about measuring lengths and angles; it's also about understanding the visual harmony of shapes around us, from famous buildings to simple letters of the alphabet.

Understanding symmetry is essential because it forms the foundation for later topics like transformations (reflections and rotations) and helps you analyze the properties of polygons (like why a square behaves differently than a trapezium). Don't worry if this seems abstract—we’ll use plenty of examples to keep it clear and fun!

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1. Line Symmetry (Reflectional Symmetry) in 2D

Line Symmetry, also called reflectional symmetry, describes a shape that can be folded exactly in half so that one side perfectly matches the other.

What is a Line of Symmetry?

A Line of Symmetry (or Axis of Symmetry) is an imaginary line that divides a shape into two mirror-image halves.

How to Test for Line Symmetry:

• Analogy: Imagine you are folding a piece of paper. If the two halves match up perfectly along the fold line, that fold is a line of symmetry.
• The line of symmetry must be perpendicular to the line segment connecting any point on one side to its mirror image on the other side.

Examples of Line Symmetry in Common Shapes

The number of lines of symmetry a shape has can vary wildly. Let’s look at some key examples:

• Square: 4 lines of symmetry (2 diagonal, 2 mid-side).
• Rectangle: 2 lines of symmetry (through the middle of opposite sides).
• Equilateral Triangle: 3 lines of symmetry (from each vertex to the midpoint of the opposite side).
• Isosceles Triangle: 1 line of symmetry (vertical, through the unique vertex).
• Parallelogram: 0 lines of symmetry (unless it is a rhombus or rectangle).
• Circle: An infinite number of lines of symmetry (any diameter).
• Regular Polygon: A regular polygon with \(n\) sides has \(n\) lines of symmetry.

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2. Rotational Symmetry in 2D

Rotational Symmetry describes a shape that looks the same after being rotated (turned) less than a full 360° around a fixed point.

Order of Rotational Symmetry

The Order of Rotational Symmetry is the number of times a shape fits exactly onto its original outline during one complete turn (360°).

Key Definitions:

• Centre of Rotation: The fixed point around which the shape turns. This is usually the centre of the shape.
• Order: This is an integer greater than 1. If a shape only looks the same after a 360° turn, it has an order of 1 (and mathematically, we say it has no rotational symmetry).

Calculating the Angle of Rotation

If you know the order of rotational symmetry (\(N\)), you can calculate the smallest angle through which you need to rotate the shape for it to match its original position. This is known as the Angle of Rotational Symmetry.

Formula:

\(\text{Angle of Rotation} = \frac{360^\circ}{\text{Order of Symmetry}}\)

Example: A square has rotational symmetry order 4. The angle of rotation is \(\frac{360^\circ}{4} = 90^\circ\).

Examples of Rotational Symmetry

Let's find the order for some common shapes:

• Square: Order 4 (it fits four times in 360°).
• Rectangle: Order 2 (it only fits twice: at 180° and 360°).
• Equilateral Triangle: Order 3.
• Rhombus: Order 2.
• Isosceles Trapezium: Order 1 (no rotational symmetry, only line symmetry).
• Circle: Order infinite (it matches its position at any angle).

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3. Symmetry in Three-Dimensional (3D) Solids (Extended Content E5.4)

Hey Extended Students! When we move into the third dimension, symmetry gets a little more complex. Instead of lines and points, we look at Planes and Axes.

3.1 Plane of Symmetry (3D Reflection)

A Plane of Symmetry is a flat surface (like a sheet of glass) that divides a 3D solid into two halves that are exact mirror images of each other.

Analogy: Imagine slicing a piece of fruit perfectly down the middle so that both halves are identical. The knife blade represents the plane of symmetry.

Examples of Planes of Symmetry:

• Cuboid (Rectangular Prism): Has 3 planes of symmetry (cutting it parallel to each pair of faces).
• Cube: Has 9 planes of symmetry. (3 parallel to faces, 6 diagonal cuts).
• Cylinder: Has an infinite number of planes of symmetry!
  (Think: One plane slicing it horizontally through the centre, and infinite vertical planes slicing through the diameter.)
• Square-based Pyramid: Has 4 planes of symmetry (slicing from the apex down through the midpoints of the base sides or diagonally through the base corners).

3.2 Axis of Rotational Symmetry (3D Rotation)

An Axis of Symmetry (in 3D) is a line around which the solid can be rotated to map exactly onto itself. The order of the axis is the number of times this happens in 360°.

Examples of Axes of Symmetry:

• Cube: Has multiple axes with different orders:
  • 3 axes passing through the centres of opposite faces (Order 4).
  • 4 axes passing through opposite vertices (Order 3).
  • 6 axes passing through the midpoints of opposite edges (Order 2).
• Cylinder:
  • 1 axis passing through the centres of the circular faces (Order Infinite).
  • Infinite axes passing through the midpoint of the central axis, perpendicular to it (Order 2).
• Cone: Has 1 axis of symmetry (the central height line) with Order Infinite, because you can rotate it by any angle and it looks the same.

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Summary of Symmetry Properties of Triangles and Quadrilaterals (Review C5.4/E5.4)

The syllabus requires you to relate the properties of polygons directly to their symmetries. Here’s a quick summary table:

Shape	Lines of Symmetry	Order of Rotational Symmetry
Equilateral Triangle	3	3
Isosceles Triangle	1	1
Square	4	4
Rectangle	2	2
Rhombus	2	2
Kite	1	1
Parallelogram	0	2
Trapezium (general)	0	1
Isosceles Trapezium	1	1

Keep practicing visualizing these transformations. Geometry is often about seeing the problem clearly, and mastering symmetry will greatly help your confidence in future topics like transformations and properties of polygons!