Final answer:
The student's question relates to geometric transformations such as rotations, translations, and reflections. It entails calculating the new coordinates after performing the specified transformations in sequence.
Step-by-step explanation:
The question involves understanding the effects of geometric transformations on an object's coordinates, specifically rotations, translations, and reflections. When an object is rotated around the origin, its new coordinates (x', y') are related to the original coordinates (x, y) through the trigonometric functions of the angle of rotation. For a 180° rotation, the new coordinates are (-x, -y), and for a 90° rotation the new coordinates are (-y, x) if the rotation is counterclockwise.
For composition A), a 180° rotation followed by a 90° rotation will result in the coordinates (-y, -x). In composition B), a 180° rotation will produce (-x, -y), and then the translation adds 5 to the x-coordinate and 2 to the y-coordinate, resulting in (-x + 5, -y + 2). Composition C) involves a reflection over the x-axis that inverts the y-coordinate, producing (x, -y), and a subsequent 90° rotation will change these to (y, x). Finally, for composition D), after the reflection (x, -y), the translation will result in (x + 5, -y + 2).
These transformations have practical applications in various fields, including physics, engineering, and computer graphics, where they are used to describe rigid-body rotations and other movements in space.