Symmetry | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Introduction

Symmetry in mathematics can be of two types:

• Line (or Plane) symmetry deals with reflections and mirror images of shapes in both 2D and 3D.
• Rotational symmetry deals with how often a shape looks identical (congruent) when it has been rotated.

Rotational Symmetry

• Rotational symmetry pertains to how many times a shape appears identical as it's rotated 360° around its center.
• This quantity is termed the order of rotational symmetry.
• Utilizing tracing paper can facilitate determining the order of rotational symmetry.
• Create an arrow on the tracing paper to aid in recognizing when the rotation completes a full 360° turn.

• When a shape returns to its original form after a rotation, it adds 1 to the order of rotational symmetry.
  • Consequently, a shape cannot have an order of 0, as it always retains its original form after a full rotation.
  • A shape characterized by rotational symmetry order 1 can be described as lacking any rotational symmetry.
  • In essence, the shape only appears identical to its original form once the rotation completes a full circle back to the starting point.

Lines of Symmetry

• Line Symmetry: Line symmetry involves shapes that can incorporate mirror lines. 
  • Each side of the line of symmetry mirrors the other side.
• Reflection: Lines of symmetry can be likened to folding lines. 
  • Folding a shape along a line of symmetry results in both parts aligning perfectly.

• Examining shapes from various angles can aid in understanding. Try turning the page for a different perspective.

• Symmetry in Shapes
  • Some questions provide a shape and a line of symmetry. 
  • Complete the shape by being cautious with diagonal lines of symmetry. Use tracing paper to trace the shape and reflection line, then flip to observe the reflection.
  • Be mindful of diagonal lines of symmetry. Use tracing paper to trace the shape and reflection line, then flip to see the reflection.
• Two-Way Reflections
  • "Two-way" reflections occur when the line of symmetry passes through the shape.

How do I solve problems involving symmetry?

• Symmetry aids in solving missing length and angle problems.

• It's helpful to draw a diagram and add lines of symmetry or supplement a given diagram with lines of symmetry. Utilize tracing paper provided during exams for assistance.
• Tracing paper can be used to enhance problem-solving by aiding in understanding symmetry.

Planes of Symmetry

What is a plane of symmetry?

• A plane represents a flat surface that can take on any 2D shape.
• A plane of symmetry is a flat plane that divides a 3D shape into two identical halves.
• If a 3D shape possesses a plane of symmetry, it also exhibits reflection symmetry, where the two halves are mirror images of each other.
• All prisms possess at least one plane of symmetry.
  • Cubes boast 9 planes of symmetry.
  • Cuboids feature 3 planes of symmetry.
  • Cylinders possess an infinite number of planes of symmetry.
  • The quantity of planes of symmetry in other prisms corresponds to the number of lines of symmetry in their cross-section plus 1.
• Pyramids can also have planes of symmetry.
• The number of planes of symmetry in other pyramids aligns with the number of lines of symmetry in their 2D base.
• If the base of the pyramid is a regular polygon with n sides, it will have n planes of symmetry.