Question

abcd is a rectangle. trapezoid aefb is congruent to trapezoid cfed. point g is the midpoint of segment ef. select all the ways we could describe the rigid transformation that takes aefb to cfed. a. reflect aefb across line ef. b. rotate aefb 180 degrees counterclockwise around point g. c. rotate aefb 180 degrees clockwise around point g. d. translate aefb by the directed line segment from f to e. then reflect across line fe. e. translate aefb by the directed line segment from f to e. then rotate 180 clockwise around point e. regular hexagon abcdef is inscribed in a circle with center h. a. what is the image of segment bc after a 180° clockwise rotation about point h? what is the image of segment c after a reflection over line

Explanation:

Step1: Analyze option A

Reflecting $AEFB$ across line $EF$ will not map $AEFB$ to $CFED$ because the orientation of the trapezoids will not match up as required. So option A is incorrect.

Step2: Analyze option B

A 180 - degree counter - clockwise rotation of $AEFB$ around point $G$ will map $AEFB$ to $CFED$ since point $G$ is the mid - point of $EF$ and the trapezoids are congruent. The corresponding points will be in the correct positions.

Step3: Analyze option C

A 180 - degree clockwise rotation of $AEFB$ around point $G$ will also map $AEFB$ to $CFED$ for the same reason as in option B. The rotation around the mid - point of the common side $EF$ will align the two congruent trapezoids.

Step4: Analyze option D

Translating $AEFB$ by the directed line segment from $F$ to $E$ and then reflecting across line $FE$ will not map $AEFB$ to $CFED$ as the sequence of these two transformations will not result in the correct alignment of the trapezoids.

Step5: Analyze option E

Translating $AEFB$ by the directed line segment from $F$ to $E$ and then rotating 180 degrees clockwise around point $E$ will not map $AEFB$ to $CFED$ as the combination of these transformations will not place the trapezoids in the correct relative positions.

For the hexagon part:
A 180 - degree clockwise rotation about the center $H$ of a regular hexagon will map segment $BC$ to segment $EF$. This is because in a regular hexagon, a 180 - degree rotation about the center maps each vertex to the vertex opposite to it.

Answer:

For the trapezoid part: B. Rotate $AEFB$ 180 degrees counterclockwise around point $G$, C. Rotate $AEFB$ 180 degrees clockwise around point $G$.
For the hexagon part: The image of segment $BC$ after a 180° clockwise rotation about point $H$ is segment $EF$.