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, and then reflect across line fe
e translate aefb by the directed line segment from f to e, and then rotate 180° clockwise around point e

Explanation:

• Option A: Reflecting \(AEFB\) across line \(EF\) will map \(AEFB\) to \(CFED\) since reflection is a rigid transformation and \(AEFB\cong CFED\).
• Option B: Rotating \(AEFB\) \(180^{\circ}\) counter - clockwise around point \(G\) (the mid - point of \(EF\)). A \(180^{\circ}\) rotation about the mid - point of a segment joining corresponding points of congruent figures (since \(AEFB\cong CFED\) and \(G\) is the mid - point of \(EF\)) is a valid rigid transformation.
• Option C: Rotating \(AEFB\) \(180^{\circ}\) clockwise around point \(G\). A \(180^{\circ}\) clockwise rotation is equivalent to a \(180^{\circ}\) counter - clockwise rotation (because \(180^{\circ}\) clockwise \(=360 - 180^{\circ}\) counter - clockwise, and for a \(180^{\circ}\) rotation, the direction does not matter in terms of the final position of the figure).
• Option D: Translating \(AEFB\) by the directed segment from \(F\) to \(E\) moves \(F\) to \(E\). Then reflecting across \(FE\) (which is the same as reflecting across \(EF\)) would not be a correct sequence. If we translate \(AEFB\) by \(\overrightarrow{FE}\), and then reflect across \(FE\), it is not a valid transformation to get \(CFED\) from \(AEFB\).
• Option E: Translating \(AEFB\) by the directed segment from \(F\) to \(E\) moves \(F\) to \(E\). Rotating \(180^{\circ}\) clockwise around \(E\) does not map \(AEFB\) to \(CFED\) as the orientation and position will not match the congruent trapezoid \(CFED\).

Answer:

A. Reflect \(AEFB\) across line \(EF\), B. Rotate \(AEFB\) \(180\) degrees counterclockwise around point \(G\), C. Rotate \(AEFB\) \(180\) degrees clockwise around point \(G\)