Geometry / Unit 4: Transformations
4.6 Symmetry
Notes
A figure has symmetry if a transformation maps it onto itself.
- Line (reflectional) symmetry: a reflection over a line maps the figure onto itself.
- Rotational symmetry: a rotation of less than $360^\circ$ maps it onto itself.
- A regular $n$-gon has $n$ lines of symmetry and rotational symmetry in multiples of $\displaystyle \frac{360^\circ}{n}$.
- Example: a regular hexagon has 6 lines of symmetry and maps onto itself by $60^\circ$ turns.
Practice
How many lines of symmetry does a square have?
Two diagonals plus two lines through the midpoints of opposite sides.
What is the smallest rotation that maps a regular pentagon onto itself?
$360^\circ \div 5 = 72^\circ$.
Which figure has rotational symmetry but no line symmetry?
A general parallelogram maps onto itself by $180^\circ$ but no reflection maps it onto itself.
Ready for more? Take the Unit 4 test.