Question

which rule describes the composition of transformations that maps figure abcde to figure a\b\cd\e\?
○ ( 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 composition of transformations notation

The notation \(r_{m}\circ R_{C,90^{\circ}}\) means first perform the rotation \(R_{C,90^{\circ}}\) (rotation about point \(C\) by \(90^{\circ}\)) and then perform the reflection \(r_{m}\) (reflection over line \(m\)). The notation \(R_{C,90^{\circ}}\circ r_{m}\) means first perform the reflection \(r_{m}\) and then perform the rotation \(R_{C,90^{\circ}}\). Similarly for \(180^{\circ}\) rotations.

Step2: Analyze the transformation order

If we first rotate the pentagon \(ABCDE\) about point \(C\) by \(180^{\circ}\) (\(R_{C,180^{\circ}}\)), and then reflect it over line \(m\) (\(r_{m}\)), we can map figure \(ABCDE\) to figure \(A''B''C''D''E''\).
Let's assume a general point \(P(x,y)\) in the original figure. A rotation \(R_{C,180^{\circ}}\) about a point \(C\) (using the rotation formula \((x - h,y - k)\to(-(x - h),-(y - k))\) if \(C=(h,k)\)) and then a reflection \(r_{m}\) (using the reflection formula over a line \(m\) which can be of the form \(y = ax + b\) or \(x = c\) etc., but geometrically by visual inspection of the figure, the order of rotation first then reflection does not match.
If we first reflect \(ABCDE\) over line \(m\) (\(r_{m}\)) and then rotate about \(C\) by \(180^{\circ}\) (\(R_{C,180^{\circ}}\)), we can match the vertices. For example, consider the orientation and position of the vertices. The reflection flips the figure over line \(m\) and then the \(180^{\circ}\) rotation about \(C\) (center - point transformation) gives the final image.

Answer:

\(r_{m}\circ R_{C,180^{\circ}}\)