Symmetry
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Line and Rotational Symmetry in 2D Shapes
A 2D shape has line symmetry if it can be reflected onto itself along a line, and rotational symmetry if it can be rotated onto itself about its centre before completing a full turn.
- A line of symmetry divides a shape into two mirror-image halves - folding along it would make the two halves match exactly
- The order of rotational symmetry is the number of positions (including the starting position) that look identical during one full 360° turn
- A shape with rotational symmetry of order 1 has no rotational symmetry (it only matches itself in its starting position)
- A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n
Worked Example: Finding the Symmetry of a Parallelogram
- Question: State the number of lines of symmetry, and the order of rotational symmetry, of a (non-rectangular) parallelogram.
- Step 1: No line reflects the parallelogram onto itself, so it has 0 lines of symmetry
- Step 2: Rotating 180° about its centre maps it onto itself, and no smaller rotation does
- Answer: 0 lines of symmetry; rotational symmetry of order 2
Worked Example: Finding the Lines of Symmetry of a Regular Polygon
- Question: State the number of lines of symmetry of a regular pentagon.
- Step 1: A regular pentagon has 5 equal sides and 5 equal angles
- Step 2: Each line of symmetry joins one vertex to the midpoint of the opposite side, passing through the centre
- Answer: 5 lines of symmetry
Common Mistakes
MistakeAssuming every parallelogram has line symmetry, just because a rectangle does
Fixonly special parallelograms (rectangle, rhombus, square) have line symmetry - a general parallelogram has none
MistakeNot counting the starting position when finding the order of rotational symmetry, giving an answer one too low
Fixthe order of rotational symmetry always includes the original starting position in its count
Symmetry Properties of 3D Solids
Three-dimensional solids can have planes of symmetry, similar to how 2D shapes have lines of symmetry. A plane of symmetry divides a solid into two mirror-image halves.
- A plane of symmetry is a flat surface that divides a solid into two halves that are exact mirror images of each other
- A prism with a regular n-sided cross-section has n vertical planes of symmetry (one through each vertex and the midpoint of the opposite side) plus 1 horizontal plane bisecting its length, giving n + 1 planes in total
- A cylinder has an infinite number of vertical planes of symmetry (through its central axis) plus 1 horizontal plane
- A cone has an infinite number of planes of symmetry, all passing through its axis
Worked Example: Finding the Cross-Section from the Number of Planes of Symmetry
- Question: A prism has 8 planes of symmetry. Its cross-section is a regular polygon. Name this polygon.
- Step 1: Total planes of symmetry = n + 1, where n is the number of sides of the cross-section
- Step 2: n+1=8 n=7
- Answer: heptagon
Common Mistakes
MistakeForgetting the extra horizontal plane of symmetry that bisects a prism's length, and using n instead of n + 1
Fixa regular prism has one extra horizontal plane in addition to the n vertical ones - always add 1 to the number of sides
MistakeConfusing a line of symmetry (a 2D idea) with a plane of symmetry (the 3D equivalent)
Fixa plane of symmetry is a flat cutting surface through a solid, not a line - think of it as slicing the solid into two identical mirror-image halves
Symmetry Properties of Circles
Extended Only
A circle's own line symmetry leads to several useful geometric facts about chords and tangents, all following from the fact that every diameter is a line of symmetry of the circle.
- Equal chords are always equidistant from the centre of the circle
- The perpendicular bisector of any chord always passes through the centre of the circle
- Tangents drawn from the same external point to a circle are always equal in length
Worked Example: Using the Equal-Tangents Property
- Question: Two tangents are drawn from an external point T to a circle, touching the circle at points A and B. Given that TA = 9 cm, find the length of TB.
- Step 1: Tangents drawn from the same external point are always equal in length
- Answer: TB = 9 cm
Common Mistakes
MistakeAssuming tangents from two different external points must also be equal
Fixthe equal-tangents rule only applies to two tangents drawn from the very same external point
MistakeForgetting that the perpendicular bisector of a chord passes through the centre, and trying to locate the centre by another method entirely
Fixthis property gives a direct way to find a circle's centre - construct the perpendicular bisectors of two different chords, and their intersection is the centre
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