Read it again any time, including while you work through the exercises.
You already know mirror (reflection) symmetry. Here are two new transformations for your geometry toolkit: point symmetry and translation.
Point symmetry is a half turn. You choose a point O, the centre, and rotate the whole figure half a turn (180°) around it. Concretely, to find the image M' of a point M: draw the line through M and O, then copy the distance OM on the other side. The centre O sits exactly at the midpoint of segment [MM']. Fun result: the image figure is upside down, like a flipped playing card.
A translation does not turn anything: it is a slide. The whole figure moves in the same direction, the same way, by the same length, like a token sliding across a board without spinning. On a grid it is easy to describe: for example, 3 squares right and 1 square up, for every point at once.
These two transformations share something with reflection: they preserve lengths, angles and areas. The image is a perfect copy of the original. What changes is its position: turned 180° for point symmetry, simply shifted for translation.
O is a point and M a point 4 cm from O. Construct M', the image of M under point symmetry with centre O.
- 1
tracer la droite (MO)
I draw the line through M and the centre O, and extend it beyond O.
- 2
OM' = OM = 4 cm
I copy the distance: I place M' on this line, on the other side of O, exactly 4 cm from O.
- 3
O = milieu de [MM']
I check: O is exactly the midpoint of segment [MM']. That is the signature of point symmetry, the perfect half turn.
Result: M' lies on line (MO), on the other side of O, with OM' = 4 cm.
On a grid, point A is at (2, 3). We apply the translation 3 squares right and 1 up. Where does A land?
- 1
2 + 3 = 5
The slide to the right adds 3 to the column: A moves from column 2 to column 5.
- 2
3 + 1 = 4
The upward slide adds 1 to the row: A moves from row 3 to row 4. And every other point of the figure slides exactly the same way!
Result: A lands at (5, 4).