Question

here are two polygons. select all sequences of translations, rotations, and reflections below that would take polygon p to polygon q. a rotate 180° around point a. b rotate 60° counterclockwise around point a and then reflect over the line fa. c translate so that a is taken to j. then reflect over line ba. d reflect over line ba and then translate by directed line segment ba. e reflect over the line ba and then rotate 60° counterclockwise around point a.

Explanation:

Step1: Analyze option A

A 180 - degree rotation around point A can map polygon P to polygon Q as it will flip the polygon in a way that the corresponding vertices match.

Step2: Analyze option B

A 60 - degree counter - clockwise rotation around point A followed by a reflection over line FA will not map polygon P to polygon Q as the orientation and position will not match the target polygon.

Step3: Analyze option C

Translating so that A is taken to J and then reflecting over line BA will not map polygon P to polygon Q as the initial translation changes the position in a non - matching way.

Step4: Analyze option D

Reflecting over line BA and then translating by directed line segment BA will map polygon P to polygon Q. The reflection flips the polygon and the translation moves it to the correct position.

Step5: Analyze option E

Reflecting over line BA and then rotating 60 degrees counter - clockwise around point A will not map polygon P to polygon Q as the rotation after reflection will distort the mapping.

Answer:

A. Rotate 180° around point A.
D. Reflect over line BA and then translate by directed line segment BA.