Question

which rule describes the composition of transformations that maps figure pqrs to figure pqrs? options: ( r_l circ r_{q, 180^circ} ), ( r_{s, 270^circ} circ r_{q, 180^circ} ), ( r circ r_l ), ( r_{q, 180^circ} circ r_{s, 270^circ} ) (diagram shows parallelograms pqrs, pqrs, pqrs with reflection line ( l ) and rotation centers q, s)

Explanation:

To solve this, we analyze the transformation composition:

Step 1: Understand Transformations

• \( r_l \): Reflection over line \( l \).
• \( R_{Q, 180^\circ} \): Rotation 180° about point \( Q \).
• \( R_{S', 270^\circ} \): Rotation 270° about point \( S' \).

Step 2: Analyze the Mapping

First, map \( PQRS \) to \( P'Q'R'S' \) (likely a rotation or reflection), then to \( P''Q''R''S'' \). The key is the composition order: \( r_l \circ R_{Q, 180^\circ} \) means "first rotate 180° about \( Q \), then reflect over \( l \)".

Step 3: Match the Rule

By examining the figure, the composition that maps \( PQRS \) to \( P''Q''R''S'' \) is \( \boldsymbol{r_l \circ R_{Q, 180^\circ}} \), as it aligns with the sequence of rotation (180° about \( Q \)) followed by reflection over \( l \).

Answer:

\( r_l \circ R_{Q, 180^\circ} \)