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: Analyze rotation

First, observe that a $180^{\circ}$ rotation ($R_{C',180^{\circ}}$) about a point $C'$ changes the orientation of the figure.

Step2: Analyze reflection

Then, a reflection ($r_{m}$) across line $m$ further modifies the position to match figure $A''B''C''D''E''$. The notation $r_{m}\circ R_{C',180^{\circ}}$ means first perform the rotation $R_{C',180^{\circ}}$ and then the reflection $r_{m}$, which is consistent with the transformation from figure $ABCDE$ to figure $A''B''C''D''E''$.

Answer:

D. $r_{m}\circ R_{C',180^{\circ}}$