2.3 Combining rotations and reflections
This week’s activities have shown that combining two rotations results in a rotation, as does combining two reflections. But what about both at once? Let’s try an example.
Activity 3 Rotating and reflecting
Starting again from the original position, rotate the triangle anticlockwise by 120 degrees, and then reflect in the line through the top corner. What happens?
Answer
The rotation moves the edges 1 → 2, 2 → 3, 3 → 1, and the reflection then swaps the two lower corners, keeping the top fixed, which means that 1 → 2 → 1, 2 → 3 → 3 and 3 → 1 → 2.
The net effect is that corner C1 is back in its initial position, while C2 and C3 have swapped places.
What single transformation is this equivalent to?
Answer
This is a reflection with respect to the symmetry line that goes through the corner in position P1.
In this case, combining a rotation with a reflection resulted in a reflection. You can check that this is always true, irrespective of your choice of rotation and reflection, and the order in which you perform them. Together, these transformations form a group.
Looking at all the possible rotations and reflections, you will find that there are only six different transformations in total. You can confirm this mathematically: you’re moving the triangle’s three corners to three available positions, so there are three choices of position for the first corner, two for the second, and just one for the last one, so in total there are 3 x 2 x 1 = 6 possible moves. Any of these six possibilities represents a possible rotation or reflection of the triangle. This includes the transformation that keeps everything in place.