Question

△abc will undergo two transformations to give △abc. which pair of transformations will give a different image of △abc if the order of the transformations is reversed?
o a. a rotation 90° counterclockwise about the origin followed by a reflection across the y - axis
o b. a translation 5 units down followed by a translation 4 units to the right
o c. a reflection across the x - axis followed by a reflection across the y - axis
o d. a rotation 180° clockwise about the origin followed by a reflection across the y - axis

Explanation:

• Option A: A rotation and a reflection are non - commutative (order matters). Rotating 90° counterclockwise then reflecting over y - axis will give a different result than reflecting over y - axis then rotating 90° counterclockwise.
• Option B: Translations are commutative. Translating 5 units down then 4 units right is the same as translating 4 units right then 5 units down. The result is a total translation of (4, - 5).
• Option C: Reflecting over x - axis then y - axis is equivalent to a 180° rotation about the origin, and reflecting over y - axis then x - axis is also a 180° rotation about the origin. So order does not matter.
• Option D: A 180° rotation about the origin is commutative with a reflection over y - axis in terms of the final image (since 180° rotation is symmetric and reflection over y - axis combined with 180° rotation will have the same result regardless of order in terms of the overall transformation effect on the triangle's position).

Answer:

A. a rotation 90° counterclockwise about the origin followed by a reflection across the y - axis