Question

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

Explanation:

Step1: Understand transformation order

In composition of transformations \(f\circ g\) means first apply \(g\) then apply \(f\).

Step2: Analyze rotation and reflection

• A rotation \(R_{C,90^{\circ}}\) (rotation about point \(C\) by \(90^{\circ}\)) and then a reflection \(r_m\) (reflection over line \(m\)) does not match the figure.
• A rotation \(R_{C,180^{\circ}}\) (rotation about point \(C\) by \(180^{\circ}\)) and then a reflection \(r_m\) does not match.
• First, perform a reflection \(r_m\) (reflection over line \(m\)) and then a rotation \(R_{C,90^{\circ}}\) (rotation about point \(C\) by \(90^{\circ}\)) does not match.
• First, perform a rotation \(R_{C,180^{\circ}}\) (rotation about point \(C\) by \(180^{\circ}\)) and then a reflection \(r_m\) (reflection over line \(m\)) does not match. Wait, no:

Let's re - check.
The correct order is first rotate the figure \(ABCDE\) about point \(C\) by \(180^{\circ}\) (\(R_{C,180^{\circ}}\)) and then reflect it over line \(m\) (\(r_m\)). In function - composition notation, when we have a composition of transformations \(T_1\circ T_2\), we apply \(T_2\) first and then \(T_1\). So \(r_m\circ R_{C,180^{\circ}}\) means first rotate \(180^{\circ}\) about \(C\) (\(R_{C,180^{\circ}}\)) and then reflect over \(m\) (\(r_m\)).

Answer:

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