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Cambridge IGCSE0607

Symmetry

International Mathematics 0607 Chapter Notes

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Symmetry
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Symmetry notes

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In this chapter

  1. 1Line symmetry
  2. 2Rotational symmetry
  3. 3Triangles, quadrilaterals and polygons
  4. 4Symmetry of solids

1. Line symmetry

A shape has line symmetry if a line can be drawn through it so that one side is the mirror image of the other. The line is a line of symmetry.

Count only lines that fold the shape exactly onto itself. A diagonal is not a line of symmetry just because it joins two corners.

Draw lines with a pencil and a straight edge.

Examiner insight

“The main error in part (d) was for candidates to include one or both diagonals of the rectangle as lines of symmetry.” “Candidates must be encouraged to use a pencil and straight edge when the answer is a diagram.”

Worked example 12 marks

A rectangle is 8 cm long and 5 cm wide. It is not a square. Write down the number of lines of symmetry of the rectangle and say where they are.

  1. 1

    The rectangle has 2 lines of symmetry. [1]

  2. 2

    Each joins the mid-points of opposite sides. The diagonals are not lines of symmetry. [1]

2. Rotational symmetry

A shape has rotational symmetry if it fits onto itself when turned about its centre through less than a full turn.

The order of rotational symmetry is the number of times the shape fits onto itself in one full turn. A shape that only fits after a full turn has order 1.

The order is not the number of lines of symmetry. Work the two out separately.

order = (360^(°))/(smallest turn that fits)

Examiner insight

“The majority of candidates scored the mark for identifying the correct rotational symmetry as 2 but many did not know the number of lines of symmetry of a parallelogram.” “Many confused rotational symmetry with line symmetry and drew a shape with line symmetry.”

Worked example 12 marks

This is a parallelogram. (a)(i) Write down the number of lines of symmetry of the parallelogram.(ii) Write down the order of rotational symmetry of the parallelogram.

  1. 1

    Number of lines of symmetry = 0 [1]

  2. 2

    Order of rotational symmetry = 2 [1]

3. Triangles, quadrilaterals and polygons

The symmetries of a shape follow from its properties. An equilateral triangle has 3 lines of symmetry and order 3. An isosceles triangle has 1 line and order 1.

A rectangle and a rhombus each have 2 lines of symmetry and order 2. A square has 4 lines and order 4. A parallelogram has 0 lines and order 2. A kite and an isosceles trapezium each have 1 line and order 1.

A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n. A regular pentagon has five lines, each through a vertex and the mid-point of the opposite side.

Examiner insight

“The most common errors seen were to only draw one vertical line of symmetry or to draw only 3 or 4 lines of symmetry.” “Some incorrectly assumed that the triangle was equilateral and spoiled their answer by drawing in extra lines.”

Worked example 11 mark

A quadrilateral has one line of symmetry and no rotational symmetry. Write down the mathematical name of this quadrilateral.

  1. 1

    Kite or isosceles trapezium [1]

4. Symmetry of solids

A plane of symmetry cuts a solid into two halves that are mirror images. An axis of symmetry is a line through a solid such that the solid fits onto itself when turned about it through less than a full turn.

In a right prism, each line of symmetry of the cross-section gives a plane of symmetry along the length. One more plane cuts the prism halfway along its length. A cuboid with three different edge lengths has 3 planes of symmetry.

A right pyramid with a square base has 4 planes of symmetry and one axis of symmetry, through the apex and the centre of the base. A cylinder has an axis of symmetry along its length, through the centres of its circular ends. A cone has one axis, through its apex and the centre of its base.

Worked example 13 marks

A right pyramid has a square base.(a) Write down the number of planes of symmetry of the pyramid.(b) Describe the axis of symmetry and give the order of rotational symmetry about it.

  1. 1

    4 planes of symmetry [1]

  2. 2

    The axis goes through the apex and the centre of the base [1]

  3. 3

    Order of rotational symmetry = 4 [1]

Recap

  • A line of symmetry folds a shape exactly onto itself. Diagonals are not automatically lines of symmetry.
  • Order of rotational symmetry = the number of times the shape fits onto itself in one full turn.
  • Parallelogram: 0 lines of symmetry, order 2.
  • Rectangle and rhombus: 2 lines, order 2. Square: 4 lines, order 4.
  • A regular polygon with n sides has n lines and order n.
  • Extended: a plane of symmetry cuts a solid into two mirror-image halves. A right prism has one for each line of symmetry of its cross-section, and one more halfway along.
  • Extended: an axis of symmetry is a line about which the solid fits onto itself when turned.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the number of lines of symmetry of a rectangle that is not a square.1 mark
  2. Write down the order of rotational symmetry of a parallelogram that is not a rectangle or a rhombus.1 mark
  3. Do not use a calculator. Write down the number of lines of symmetry and the order of rotational symmetry of an equilateral triangle.2 marks
  4. A quadrilateral has 2 lines of symmetry and rotational symmetry of order 2. Its sides are not all equal. Write down its mathematical name.1 mark
  5. Do not use a calculator. Write down the number of lines of symmetry and the order of rotational symmetry of a regular hexagon.2 marks
  6. Do not use a calculator. A pattern fits onto itself for the first time after a turn of 72^(°) about its centre. Work out the order of rotational symmetry.2 marks
  7. A cuboid has three different edge lengths. Write down the number of planes of symmetry of the cuboid.1 mark
  8. A right prism has regular hexagons as its two ends. Write down the number of planes of symmetry of the prism and the order of rotational symmetry about the axis along its length.3 marks
  9. A wooden door wedge is a right prism. Its cross-section is a parallelogram with sides 6 cm and 4 cm and no right angles. (a) Work out the number of planes of symmetry of the wedge. (b) Write down the order of rotational symmetry about the axis along its length.3 marks
  10. Two tent designs are right pyramids, each with its apex directly above the centre of its base. Tent P has a rectangular base 8 m by 5 m. Tent Q has a rhombus as its base, with diagonals 8 m and 5 m. (a) For each tent, write down the number of planes of symmetry and describe where these planes meet the base. (b) Write down the order of rotational symmetry of each tent about the axis through its apex.5 marks

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