Central symmetry and translation
Translations
Definition
A translation moves all the points of a figure in the same direction, along the same line, and by the same distance. It is said to be defined by a vector, denoted, for example, by vector AB.
If A’ is the image of A under the translation by vector AB, then the path from A to A’ is parallel to AB, of the same length and in the same direction.
Practical example: sliding a stamp across a sheet of paper without turning or flipping it is an example of a translation. A wallpaper pattern that repeats regularly is created by successive translations.
Constructing the image of a point by translation
To construct A’, the image of A under the translation that maps C to D (vector CD):
- Draw a line parallel to (CD) passing through A.
- Mark off on this parallel line a distance equal to CD, in the same direction.
On a grid, this is very simple: count the horizontal and vertical displacement between C and D (for example, ‘3 squares to the right and 2 squares up’), then apply exactly the same displacement to every point in the figure.
Properties preserved
- Lengths, angles and alignment are preserved (as with symmetries).
- Furthermore, translation preserves parallelism and orientation: the figure is neither flipped nor rotated, only moved.
Common pitfall
Be careful with the direction of the movement: ‘3 squares to the right’ is not the same translation as ‘3 squares to the left’, even if the distance is the same. You must always take into account the direction, sense and length specified by the vector.