Question

which rule describes the composition of transformations that maps figure abcde to figure abcde?
r_{c,90^{circ}}circ r_{m}
r_{m}circ r_{c,90^{circ}}
r_{c,180^{circ}}circ r_{m}
r_{m}circ r_{c,180^{circ}}

Explanation:

Step1: Understand transformation order

In composition of transformations \(T_1\circ T_2\), we first apply \(T_2\) and then \(T_1\).

Step2: Analyze reflection and rotation

• A reflection \(r_m\) (over line \(m\)) is a transformation that flips a figure over a line. A rotation \(R_{C',\theta}\) rotates a figure about a point \(C'\) by an angle \(\theta\).
• If we first rotate the original figure \(ABCDE\) about \(C'\) by \(180^{\circ}\) (\(R_{C', 180^{\circ}}\)) and then reflect (\(r_m\)) the rotated figure, we can observe the mapping from \(ABCDE\) to \(A''B''C''D''E''\).

Answer:

\(r_m\circ R_{C',180^{\circ}}\) (the fourth option)