Central symmetry and translation
Central symmetry
Definition
Central symmetry transforms a figure about a point, known as the centre of symmetry, often denoted by O. The image of a point A under central symmetry about O is the point A' such that O is the midpoint of the segment [AA'].
In other words, A, O and A’ lie on a straight line, and OA = OA’.
Example: the letters H, I, N, O, S, X and Z have a centre of symmetry. The ‘+’ sign and a rectangle also have a centre of symmetry (their centre).
Constructing the symmetric point via central symmetry
To construct A’, the symmetric point of A with respect to O:
- Draw the line (AO).
- Place A’ on this line, on the opposite side of O, such that OA’ = OA.
On a grid: count the number of horizontal and vertical squares between A and O, then mark out the same number in the opposite direction from O.
Properties
| Property | Preserved? |
|---|---|
| Lengths | Yes |
| Angles | Yes |
| Direction of rotation | Yes (unlike axial symmetry) |
| Direction of lines | Yes, a line becomes a parallel line |
Common pitfall
Do not confuse central symmetry (about a point) with axial symmetry (about a line): central symmetry rotates the figure by half a turn (180 degrees) about O; it is not ‘flipped’ as in a mirror.