Geometry

Question Analysis:
This question focuses on understanding the relationship between a point, its image after a 180° rotation, and the center of rotation.
It tests knowledge of the properties of 180° rotations.

Key Concept Explanation:
When a point Q(x, y) is rotated 180° about a center of rotation, the general formula for the new coordinates Q'(x', y') is given by the mid - point formula.
If the center of rotation is the origin (0, 0), the rule for a 180° rotation is (x', y') = (-x, -y).
Also, the center of rotation is the mid - point between any point and its image after a 180° rotation.

Step - by - Step Solution:
If the center of rotation is the origin (0, 0), for a point Q(x, y), after a 180° rotation, the new point Q' has coordinates (-x, -y) according to the rotation rule.
The mid - point between Q(x, y) and Q'(-x, -y) is calculated as