Final answer:
The correct answer is that the length of segment BC is the same as segment XY after a rotation about the origin, as rotations preserve the lengths of segments.
Step-by-step explanation:
If BC is the image of XY after a rotation about the origin, the property that always remains true is the length of the segments. Rotation is a transformation that turns a figure around a fixed point called the center of rotation.
In this case, the origin is the center of rotation. Because a rotation is an isometry, which means it preserves distances, the lengths of segments are preserved during rotation. This is why the length of segment BC will be equal to the length of segment XY, regardless of their orientation or quadrant post-rotation.
Options A and B can be eliminated because a rotation does not necessarily preserve the orientation of the segment, meaning BC could be at any angle. Option C can be dismissed because rotations can move segments from one quadrant to another. Therefore, the correct answer is D: The length of BC is the same as XY.