To determine which rule correctly describes the composition of transformations that maps the pre-image [tex]$A B C D$[/tex] to the final image [tex]$A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}$[/tex], let's analyze the given options step-by-step.
1. Option 1: [tex]\( r_{x-1xi s} \circ T_{-6,1}(x, y) \)[/tex]
- This notation suggests a sequence of transformations, but its specific meaning is ambiguous and not standard in conventional geometric transformations.
2. Option 2: [tex]\( T_{-6.1} \)[/tex] or [tex]\( r_{x-2xi 5}(x, y) \)[/tex]
- This option combines two different transformations, suggesting either a translation by vector [tex]\((-6.1)\)[/tex] or another ambiguous transformation denoted by [tex]\( r_{x-2xi 5}(x, y) \)[/tex]. The meaning and application of this transformation are unclear and not standard.
3. Option 3: [tex]\( R_{0,90^{\circ}} \circ T_{-6,1}(x, y) \)[/tex]
- This rule represents a sequence of two transformations:
1. [tex]\( T_{-6,1}(x, y) \)[/tex]: Translate the figure by [tex]\((-6, 1)\)[/tex].
2. [tex]\( R_{0,90^{\circ}} \)[/tex]: Rotate the figure around the origin (0,0) by [tex]\(90^{\circ}\)[/tex] counterclockwise.
- This composition is clear and follows standard notation, first moving the figure and then rotating it.
4. Option 4: [tex]\( T_{-6,1} \circ R_{0,90}(x, V) \)[/tex]
- This rule also represents a sequence of two transformations but in reverse order:
1. [tex]\( R_{0,90}(x, V) \)[/tex]: Rotate the figure around the origin (0,0) by [tex]\(90^{\circ}\)[/tex] counterclockwise.
2. [tex]\( T_{-6,1}(x, y) \)[/tex]: Translate the figure by [tex]\((-6, 1)\)[/tex].
- This composition is also clear but involves rotating first and then translating.
Given these analyses, the correct composition of transformations that accurately maps the figure from pre-image [tex]$A B C D$[/tex] to final image [tex]$A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}$[/tex] is:
Option 3: [tex]\( R_{0,90^{\circ}} \circ T_{-6,1}(x, y) \)[/tex]
