Question

compositions
which rule describes the composition of transformations that maps figure pqrs to figure
p\q
\s\?
r_{l} \circ r_{q, 180^{\circ}} \quad r \circ r_{l}
r_{q, 180^{\circ}} \circ r_{s, 270^{\circ}} \quad r_{s, 270^{\circ}} \circ r_{q, 180^{\circ}}

Explanation:

Step1: Analyze Transformation Order

First, recall that a composition of transformations \( A \circ B \) means apply \( B \) first, then \( A \). We need to see the sequence from \( PQRS \) to \( P''Q''R''S'' \). The first transformation from \( PQRS \) (original) to \( P'Q'R'S' \) seems a rotation, then to \( P''Q''R''S'' \) a reflection over line \( l \). Wait, no—wait, the options have \( r_l \) (reflection over \( l \)) and rotations. Wait, let's check the first step: from \( PQRS \) to \( P'Q'R'S' \): rotating \( 180^\circ \) around \( Q \) (since \( Q \) is a common point? Wait, \( PQRS \) to \( P'Q'R'S' \): if we rotate \( 180^\circ \) around \( Q \), then reflect over \( l \). Wait, the composition \( r_l \circ R_{Q, 180^\circ} \) means first rotate \( 180^\circ \) around \( Q \), then reflect over \( l \). Let's verify: original \( PQRS \), rotate \( 180^\circ \) around \( Q \) to get \( P'Q'R'S' \) (matches the middle figure), then reflect over \( l \) (the horizontal line) to get \( P''Q''R''S'' \) (top figure). So the composition is reflection over \( l \) after rotation \( 180^\circ \) around \( Q \), which is \( r_l \circ R_{Q, 180^\circ} \).

Step2: Match with Options

Looking at options: first option is \( r_l \circ R_{Q, 180^\circ} \), which is apply \( R_{Q, 180^\circ} \) first, then \( r_l \). Let's check the figures: original \( PQRS \), rotate \( 180^\circ \) around \( Q \) (so \( Q \) stays, \( P \) maps to \( P' \), \( S \) to \( S' \), \( R \) to \( R' \))—that's the middle figure. Then reflect over line \( l \) (horizontal line) to get the top figure \( P''Q''R''S'' \). So the composition is \( r_l \) (reflection) after \( R_{Q, 180^\circ} \) (rotation), so \( r_l \circ R_{Q, 180^\circ} \), which is the first option.

Answer:

\( \boldsymbol{r_l \circ R_{Q, 180^\circ}} \) (the first option: \( r_l \circ R_{Q, 180^\circ} \))