Question

which rule describes the composition of transformations that maps figure pqrs to figure pqrs?
( r circ r_l )
( r_l circ r_{q, 180^circ} )
( r_{q, 180^circ} circ r_{s, 270^circ} )
( r_{s, 270^circ} circ r_{q, 180^circ} )

Explanation:

Step1: Analyze the first transformation

First, we consider a rotation of \( 180^{\circ} \) about point \( Q \) (denoted as \( R_{Q, 180^{\circ}} \)) for the original figure \( PQRS \). This rotation will map \( PQRS \) to an intermediate figure (like \( P'Q R' S' \) in the diagram).

Step2: Analyze the second transformation

Then, we apply a reflection over line \( l \) (denoted as \( r_{l} \)) to the intermediate figure. This reflection will map the intermediate figure to the final figure \( P''Q''R''S'' \). In function composition, the rightmost transformation is applied first, so the composition is \( r_{l} \circ R_{Q, 180^{\circ}} \), meaning we first do \( R_{Q, 180^{\circ}} \) and then \( r_{l} \). By checking the other options:

• \( R \circ r_{l} \): The order and transformations don't match the mapping.
• \( R_{Q, 180^{\circ}} \circ R_{S', 270^{\circ}} \) and \( R_{S', 270^{\circ}} \circ R_{Q, 180^{\circ}} \): These involve rotations about different points in an order that doesn't align with the mapping from \( PQRS \) to \( P''Q''R''S'' \) as we observe the diagram (the key is the rotation about \( Q \) first then reflection over \( l \)).

Answer:

\( r_{l} \circ R_{Q, 180^{\circ}} \)