Line and Rotational Symmetry
A shape has line symmetry if a fold line makes the two halves match, and rotational symmetry of order \(n\) if it fits onto itself \(n\) times in one full turn. This NSW Year 9 Advanced maths subtopic covers symmetry in shapes and graphs (NESA outcome MA5‑LIN‑P‑01).
You will state the order of rotational symmetry and the axes of symmetry of polygons, letters, quadrilaterals and tetrominoes, and find the smallest angle of turn. You will describe the symmetry of the graphs of \(y = x^2 - 4\), \(y = x^3\) and \(y = \frac{1}{x}\), use symmetry to find points on a parabola, complete a pattern with order \(4\), and explain why two perpendicular lines of symmetry give a half-turn. It is part of Year 9 Advanced Properties of Geometric Figures.
Theory
A shape has line symmetry when a mirror line folds it onto itself, and rotational symmetry when a turn of less than \(360^\circ\) fits it onto itself. This Year 9 Advanced maths page covers the order of rotational symmetry, axes of symmetry of regular polygons, and the symmetry of graphs such as parabolas, cubics and hyperbolas, using coordinates. It is useful if you are heading for Advanced maths.
A shape has line symmetry if it can be folded along a line so that the two halves match exactly. That line is an axis of symmetry (or line of symmetry). A shape can have none, one or several.
A shape has rotational symmetry if a turn about a fixed point, the centre of rotation, fits it exactly onto itself before a full turn. The order of rotational symmetry is the number of times it fits onto itself in one full turn of \(360^\circ\), counting the full turn. A shape that only fits after a full turn has order \(1\), which means no rotational symmetry.
A regular polygon with \(n\) sides has \(n\) axes of symmetry and rotational symmetry of order \(n\). If \(n\) is odd, each axis joins a vertex to the midpoint of the opposite side. If \(n\) is even, half the axes join opposite vertices and half join the midpoints of opposite sides.
Graphs on the number plane can be symmetric too. A parabola has one vertical axis of symmetry through its turning point. Rotational symmetry of order \(2\) about the origin is also called point symmetry: every point \((x, y)\) on the graph has a partner \((-x, -y)\).
Line symmetry and rotational symmetry are separate properties. A shape can have one without the other.
If a shape has rotational symmetry of order \(n\), the smallest turn that fits it onto itself is:
Coordinate rules for testing a graph or completing a shape:
| Symmetry | Point \((x, y)\) goes to |
|---|---|
| axis: the \(y\)-axis (\(x = 0\)) | \((-x, y)\) |
| axis: the \(x\)-axis (\(y = 0\)) | \((x, -y)\) |
| axis: the line \(y = x\) | \((y, x)\) |
| axis: the line \(y = -x\) | \((-y, -x)\) |
| half-turn about \(O\) (order \(2\)) | \((-x, -y)\) |
| quarter-turn about \(O\), anticlockwise | \((-y, x)\) |
How to find the symmetries of a shape
- Lines: try each possible fold line: through vertices, through midpoints of sides, horizontal, vertical and diagonal. Count only the ones where the halves match exactly.
- Rotation: find the centre, then the smallest turn that fits the shape onto itself. The order is \(360^\circ\) divided by that turn.
- Combined figures: a symmetry of the whole figure must fit every part onto itself, so it can only be a symmetry that the parts share.
- Patterns on a grid: for order \(4\), every shaded square must appear again after turns of \(90^\circ\), \(180^\circ\) and \(270^\circ\). Shade each square's three images, and count only squares not already shaded.
How to test a graph
- Use a rule from the Formulas table, for example \((x, y) \to (-x, -y)\) for order \(2\) about \(O\).
- Substitute into the equation and simplify, or test a few points.
- To show a symmetry is missing, one point that fails is enough. A single example that breaks a claim is called a counterexample.
(i) its number of axes of symmetry
(ii) its order of rotational symmetry
(iii) the smallest angle it can be turned through to fit onto itself.
Find the number of sides first, using the exterior angle.
| \(\text{exterior}\) | \(=\) | \(180^\circ - 150^\circ = 30^\circ\) |
| \(n\) | \(=\) | \(\dfrac{360}{30} = 12\) |
A regular polygon with \(n\) sides has \(n\) axes of symmetry and rotational symmetry of order \(n\).
(i) \(\mathbf{12}\) axes: \(6\) through opposite vertices and \(6\) through the midpoints of opposite sides.
(ii) order \(\mathbf{12}\)
| \(\text{(iii) smallest turn}\) | \(=\) | \(\dfrac{360^\circ}{12} = \mathbf{30^\circ}\) |
(i) Find the other three vertices.
(ii) Is the \(y\)-axis an axis of symmetry of the hexagon?
(i) Order \(2\) means a half-turn (\(180^\circ\)) about \(O\) fits the hexagon onto itself. A half-turn about \(O\) changes the sign of both coordinates: \((x, y) \to (-x, -y)\).
| \(A(1, 3)\) | \(\to\) | \(A'(-1, -3)\) |
| \(B(4, 1)\) | \(\to\) | \(B'(-4, -1)\) |
| \(C(2, -1)\) | \(\to\) | \(C'(-2, 1)\) |
(ii) Reflecting in the \(y\)-axis sends \((x, y) \to (-x, y)\), so \(A(1, 3)\) would go to \((-1, 3)\). That point is not a vertex, so the \(y\)-axis is not an axis of symmetry.
(i) Find the equation of its axis of symmetry.
(ii) The point \((1, -5)\) is on the parabola. Use symmetry to find another point with \(y = -5\).
(i) The axis of symmetry is halfway between the \(x\)-intercepts.
| \(x\) | \(=\) | \(\dfrac{-4 + 2}{2}\) |
| \(\) | \(=\) | \(-1\) |
The axis is the vertical line \(\boldsymbol{x = -1}\).
(ii) \((1, -5)\) is \(2\) units to the right of \(x = -1\). Its mirror image is \(2\) units to the left, at \(x = -3\), with the same \(y\)-value: \(\boldsymbol{(-3, -5)}\).
| \(\text{Check: } (-3)^2 + 2(-3) - 8\) | \(=\) | \(9 - 6 - 8 = -5\) |
A half-turn about \(O\) sends \((x, y) \to (-x, -y)\). Replace \(x\) by \(-x\) and compare with \(-y\).
| \((-x)^3 - 4(-x)\) | \(=\) | \(-x^3 + 4x\) |
| \(\) | \(=\) | \(-(x^3 - 4x)\) |
So whenever \((x, y)\) is on the graph, \((-x, -y)\) is too. For example \((1, -3) \to (-1, 3)\). The graph fits onto itself after \(180^\circ\) and \(360^\circ\): order 2.
For the \(y\)-axis, \((1, -3)\) would reflect to \((-1, -3)\). But \(x = -1\) gives \(y = -1 + 4 = 3\), not \(-3\). One point that fails is enough: the \(y\)-axis is not an axis of symmetry.
Common pitfalls
Frequently asked questions
What is the order of rotational symmetry?
It is the number of times a shape fits exactly onto itself during one full turn of \(360^\circ\), counting the full turn. A square has order \(4\) because it fits every \(90^\circ\).
How many lines of symmetry does a regular polygon have?
A regular polygon with \(n\) sides has \(n\) lines of symmetry. A regular octagon has \(8\), and it also has rotational symmetry of order \(8\).
Can a shape have rotational symmetry but no line symmetry?
Yes. A parallelogram that is not a rectangle or rhombus fits onto itself after a half-turn, so it has rotational symmetry of order \(2\), but no fold line matches its halves.
How do you find the axis of symmetry of a parabola?
Find two points on the parabola with the same \(y\)-value, such as the \(x\)-intercepts, and take the \(x\)-value halfway between them. The axis is the vertical line through that value, for example \(x = -1\).
What does point symmetry mean for a graph?
A graph has point symmetry about the origin when turning it \(180^\circ\) about the origin fits it onto itself. Algebraically, whenever \((x, y)\) is on the graph, \((-x, -y)\) is on it too.
Does a circle have rotational symmetry?
Yes. A circle fits onto itself after a turn of any angle about its centre, and every line through its centre is an axis of symmetry, so it has infinitely many of each.