Question

here are 2 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. translate so that a is taken to j. then reflect over line ba.
c. rotate 60° counterclockwise around point a and then reflect over the line fa.
d. reflect over the line ba and then rotate 60° counterclockwise around point a.
e. reflect over line ba and then translate by directed line segment ba.

Explanation:

Step1: Analyze Option A

A \(180^{\circ}\) rotation around point \(A\) will map polygon \(P\) to polygon \(Q\) as it changes the orientation and position in a way that aligns the two polygons.

Step2: Analyze Option B

Translating \(A\) to \(J\) first changes the position, but then reflecting over line \(BA\) does not map \(P\) to \(Q\) correctly.

Step3: Analyze Option C

A \(60^{\circ}\) counter - clockwise rotation around \(A\) followed by a reflection over \(FA\) does not result in \(P\) being mapped to \(Q\).

Step4: Analyze Option D

Reflecting over \(BA\) first and then a \(60^{\circ}\) counter - clockwise rotation around \(A\) does not map \(P\) to \(Q\).

Step5: Analyze Option E

Reflecting over \(BA\) and then translating by the directed line segment \(BA\) maps polygon \(P\) to polygon \(Q\) as the reflection and translation combination aligns the two polygons.

Answer:

A. Rotate \(180^{\circ}\) around point \(A\); E. Reflect over line \(BA\) and then translate by directed line segment \(BA\)