The description indicates that quadrilateral ABCD is moved to quadrilateral A'B'C'D' and that A'B'C'D' is located in the third quadrant, while ABCD is in the first quadrant. Since a reflection over the x-axis changes the sign of the y-coordinates, it would move points from the first quadrant to the fourth quadrant, not the third.
A reflection over the y-axis would change the sign of the x-coordinates, but it would not move a figure from the first quadrant to the third quadrant either.
A rotation about the origin would keep both quadrilaterals in the same type of quadrant (either both in the first or both in the third).
Since the transformation moves each point from the first quadrant to the third quadrant, the most likely transformation is a reflection over the origin (which isn't one of the given options, but you can consider it a combination of reflections over both axes). However, from the given options, the best match would be:
C. Rotation about the origin
It's important to clarify that this is a bit of an atypical scenario if strictly speaking about the options you have, as it's not an exact match to the common transformations but involves the concept of moving quadrilaterals across the axes.