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 \(EF\) would not map \(AEFB\) to \(CFED\) as the orientation and position relative to \(EF\) are not such that a simple reflection over \(EF\) would work.
• Option B: Since \(G\) is the mid - point of \(EF\), rotating \(AEFB\) \(180^{\circ}\) counter - clockwise around \(G\) (a rotation of \(180^{\circ}\) counter - clockwise is equivalent to a \(180^{\circ}\) clockwise rotation in terms of the final position of the figure) will map \(AEFB\) to \(CFED\). Because for a rotation of \(180^{\circ}\) about a point \(G\) (the mid - point), each point \(P(x,y)\) in \(AEFB\) will be mapped to a point \(P'(-x + 2x_G,-y+ 2y_G)\) (using the rotation formula about a point \((x_G,y_G)\)) and since \(AEFB\cong CFED\), this rotation works.
• Option C: Rotating \(AEFB\) \(180^{\circ}\) clockwise around \(G\) (a \(180^{\circ}\) rotation, regardless of clockwise or counter - clockwise direction when the center of rotation is a mid - point and the figures are congruent) will map \(AEFB\) to \(CFED\). The formula for a \(180^{\circ}\) rotation about a point \((x_G,y_G)\) is \((x,y)\to(2x_G - x,2y_G - y)\). Since \(AEFB\) and \(CFED\) are congruent and \(G\) is the mid - point of \(EF\), this rotation is valid.
• Option D: Translating \(AEFB\) by the directed line segment from \(F\) to \(E\) (which moves the trapezoid) and then reflecting across \(FE\) (which is the same as \(EF\)) will not map \(AEFB\) to \(CFED\) correctly. The combination of translation and reflection in this way does not account for the congruence and the position of the trapezoids.
• Option E: Translating \(AEFB\) by the directed line segment from \(F\) to \(E\) (moves the trapezoid) and then rotating \(180^{\circ}\) clockwise around \(E\). A \(180^{\circ}\) rotation about \(E\) after the translation does not map the trapezoid \(AEFB\) to \(CFED\) as the relative positions of the vertices (based on the congruence of \(AEFB\) and \(CFED\)) are not satisfied by this combination of transformation.

Answer:

B. Rotate \(AEFB\) 180 degrees counterclockwise around point \(G\), C. Rotate \(AEFB\) 180 degrees clockwise around point \(G\)