We are given two polygons, $P$ and $Q$, on a triangular grid. We need to find all sequences of transformations (translations, rotations, and reflections) that would transform polygon $P$ into polygon $Q$. The options are: A. Rotate $180^{\circ}$ around point $A$. B. Rotate $60^{\circ}$ 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 $60^{\circ}$ counterclockwise around point $A$.

1. Problem Description

We are given two polygons, $P$ and $Q$, on a triangular grid. We need to find all sequences of transformations (translations, rotations, and reflections) that would transform polygon $P$ into polygon $Q$. The options are:

A. Rotate $180^{\circ}$ around point $A$.

B. Rotate $60^{\circ}$ 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 $60^{\circ}$ counterclockwise around point $A$.

2. Solution Steps

A. Rotate $180^{\circ}$ around point $A$. This would rotate $P$ twice the angle between $AP$ and $AQ$ around $A$, which would not map $P$ onto $Q$. Thus, option A is incorrect.

B. Rotate $60^{\circ}$ counterclockwise around point $A$ and then reflect over the line $FA$. First, we rotate $P$ $60^{\circ}$ counterclockwise around $A$. Then, we reflect the resulting image over line $FA$. This sequence of transformations maps $P$ to $Q$. Thus, option B is correct.

C. Translate so that $A$ is taken to $J$. Then reflect over line $BA$. First, translate $P$ so that $A$ is translated to $J$. Then, we reflect the resulting image over line $BA$. Since the translation vector is parallel to $BA$, this would not map $P$ onto $Q$. Thus, option C is incorrect.

D. Reflect over line $BA$ and then translate by directed line segment $BA$. First, we reflect $P$ over line $BA$. Then, we translate the resulting image by directed line segment $BA$. This sequence of transformations maps $P$ to $Q$. Thus, option D is correct.

E. Reflect over the line $BA$ and then rotate $60^{\circ}$ counterclockwise around point $A$. First reflect over $BA$. This maps $A$ to $A$. Rotating the result 60 degrees counterclockwise around $A$ takes the image closer to Q, but doesn't align it. Thus, option E is incorrect.

3. Final Answer

B. Rotate $60^{\circ}$ counterclockwise around point $A$ and then reflect over the line $FA$.

D. Reflect over line $BA$ and then translate by directed line segment $BA$.