Question

Recognizing Equivalent Sequences of Transformations
Choose all sequences of transformations that produce the same image of a given figure.
a reflection across the x-axis followed by another reflection across the x-axis
a reflection across the y-axis followed by a counter-clockwise rotation 90° about the origin
a rotation of 180 degrees followed by another rotation of 180 degrees
a translation to the right followed by an equal translation to the left
a rotation 90° about the origin followed by a counter-clockwise rotation 90° about the origin

Answer 1

Answers: (a), (c), and (d).

Brief reasons:
- (a) Two reflections across the same line undo each other → identity.
- (c) 180° + 180° = 360° → identity.
- (d) A translation right followed by the equal translation left cancels → identity.

(b) is a reflection across the y-axis followed by a 90° CCW rotation (overall a reflection across y = −x), not the identity. (e) two 90° CCW rotations give a single 180° rotation, not the identity.