Symmetry (Edexcel GCSE Maths: Foundation)

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Jamie W

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Jamie W

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Symmetry

What is symmetry?

  • Symmetry can refer to:
    • Line symmetry which deals with reflections and mirror images of shapes in both 2D and 3D
    • Rotational symmetry which deals with how often a shape looks identical (congruent) when it has been rotated

Rotational Symmetry

What is the order of rotational symmetry?

  • Rotational symmetry refers to the number of times a shape looks the same as it is rotated 360° about its centre
  • This number is called the order of rotational symmetry
  • Tracing paper can help work out the order of rotational symmetry
    • Draw an arrow on the tracing paper so you can easily tell when you have turned it through 360°

finding the order of rotational symmetry using tracing paper

finding the order of rotational symmetry using tracing paper part 2finding the order of rotational symmetry using tracing paper part 3

  • Notice that returning to the original shape contributes 1 to the order
    • This means a shape can never have order 0
    • A shape with rotational symmetry order 1 may be described as not having any rotational symmetry

    • The only time it looks the same is when you get back to the start

Examiner Tip

  • Remember to use the trick above; using an upwards arrow on the tracing paper to show the starting orientation of the shape

Worked example

For the shape below, shade exactly 4 more squares so that the shape has rotational symmetry of order 4.

3-1-line-and-rotation-symmetry-we

The shape below appears the same 4 times if rotated through 360 degrees

 

3-1-1-rotation-symmetry-we-answer

Lines of Symmetry

What is line symmetry?

  • Line symmetry refers to shapes that can have mirror lines added to them
    • Each side of the line of symmetry is a reflection of the other side
  • Lines of symmetry can be thought of as a folding line too
    • Folding a shape along a line of symmetry results in the two parts sitting exactly on top of each other

Lines of symmetry in isosceles triangles, squares, and rectangles

  • It can help to look at shapes from different angles; turn the page to do this

rotating a shape to see lines of symmetry

  • Some questions will provide a portion of a shape and a line of symmetry, and you need to fill in the remaining half of the shape
  • Be careful with diagonal lines of symmetry
    • Use tracing paper to trace the shape and then flip along the line of symmetry
  • Two-wayreflections (like part c below) occur if the line of symmetry passes through the shape

 

Symm Notes fig2c (1), downloadable IGCSE & GCSE Maths revision notes Examples of reflecting in a line of symmetry

Symm Notes fig1, downloadable IGCSE & GCSE Maths revision notes

Examiner Tip

  • It can help to add the lines of symmetry to a diagram if one is given in a question
  • You should be provided with tracing paper in the exam, use this to help you
    • You can request it if you are not given it at the start

Worked example

Consider the shape below.

Symmetry worked example question

 

(a)
Write down the number of lines of symmetry.
  
The only line of symmetry is shown below

Symmetry worked example solution a

There is 1 line of symmetry.

 

(b)
Shade exactly 4 more squares so that the shape has 4 lines of symmetry.
     
The shape below has a horizontal, a vertical, and 2 diagonal lines of symmetry
  

Symmetry worked example solution b

Planes of Symmetry

What is a plane of symmetry?

  • A plane is a flat surface that can be any 2D shape
  • A plane of symmetry is a plane that splits a 3D shape into two congruent (identical) halves
  • If a 3D shape has a plane of symmetry, it has reflection symmetry
    • The two congruent halves are identical, mirror images of each other
  • All prisms have at least one plane of symmetry
    • Cubes have 9 planes of symmetry
    • Cuboids have 3 planes of symmetry
    • Cylinders have an infinite number of planes of symmetry
    • The number of planes of symmetry in other prisms will be equal to the number of lines of symmetry in its cross-section plus 1
  • Pyramids can have planes of symmetry too
    • The number of planes of symmetry in other pyramids will be equal to the number of lines of symmetry in its 2D base
    • If the base of the pyramid is a regular polygon of n sides, it will have n planes of symmetry

3-1-1-cie-igcse-planes-of-symmetry-diagram-1

Examiner Tip

If you’re unsure in the exam, consider the properties of the 3D shape.

  • Is it a prism or a pyramid?
  • How many lines of symmetry are there in the 2D faces or cross-section?

Worked example

The diagram below shows a cuboid of length 8 cm, width 5 cm and height 11 cm.

Write down the number of planes of symmetry of this cuboid.
 

cie-igcse-2020-oct-nov-p4-tz3-q6a

A plane of symmetry is where a shape can be "sliced" such that it is symmetrical.
A cuboid with three different pairs of opposite rectangles has 3 planes of symmetry.

3 planes of symmetry

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.