To determine which composite transformation accurately maps [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A^{\prime \prime} B^{\prime} C^{-}\)[/tex], we need to analyze each given transformation rule option and understand the geometric transformations involved. Let's break down each option step by step.
1. Option 1: [tex]\( r_{=} \circ R_{B^{\prime}, 90^{\circ}} \)[/tex]
- [tex]\(R_{B^{\prime}, 90^{\circ}}\)[/tex]: This denotes a 90-degree rotation around point [tex]\(B^{\prime}\)[/tex].
- [tex]\(r_{=}\)[/tex]: This denotes a reflection over some line.
This option first rotates the triangle 90 degrees around point [tex]\(B^{\prime}\)[/tex] and then reflects the resulting triangle.
2. Option 2: [tex]\( R_{B^{\prime}, 900} \circ r_m \)[/tex]
- [tex]\(r_m\)[/tex]: This denotes a reflection over some line [tex]\(m\)[/tex].
- [tex]\(R_{B^{\prime}, 900}\)[/tex]: This denotes a rotation, but the 900 degrees specified is likely a typo. Assuming it means a 90-degree rotation:
- [tex]\(R_{B^{\prime}, 90^{\circ}}\)[/tex]
This option first reflects the triangle over line [tex]\(m\)[/tex] and then rotates the resulting triangle 90 degrees around point [tex]\(B^{\prime}\)[/tex].
3. Option 3: [tex]\( r_m \circ R_{B^{\prime}, 270^{\circ}} \)[/tex]
- [tex]\(R_{B^{\prime}, 270^{\circ}}\)[/tex]: This denotes a 270-degree rotation around point [tex]\(B^{\prime}\)[/tex].
- [tex]\(r_m\)[/tex]: This denotes a reflection over some line [tex]\(m\)[/tex].
This option first rotates the triangle 270 degrees around point [tex]\(B^{\prime}\)[/tex] and then reflects the resulting triangle over line [tex]\(m\)[/tex].
4. Option 4: [tex]\( R_{B^{\prime}, 270^{\circ}} \circ r_m \)[/tex]
- [tex]\(r_m\)[/tex]: This denotes a reflection over some line [tex]\(m\)[/tex].
- [tex]\(R_{B^{\prime}, 270^{\circ}}\)[/tex]: This denotes a 270-degree rotation around point [tex]\(B^{\prime}\)[/tex].
This option first reflects the triangle over line [tex]\(m\)[/tex] and then rotates the resulting triangle 270 degrees around point [tex]\(B^{\prime}\)[/tex].
Given the above information and considering the composition of transformations:
- A 270-degree rotation is effectively a 90-degree rotation counterclockwise.
- Reflection followed by rotation can result in the required transformation when visually analyzing the steps.
Given the transformations, the combination that best fits the requirement is:
Option 3: [tex]\(r_m \circ R_{B^{\prime}, 270^{\circ}}\)[/tex]
Therefore, the rule describing the composite transformations that map [tex]\(\triangle ABC\)[/tex] to [tex]\(\triangle A^{\prime \prime} B^{\prime} C^{-}\)[/tex] is:
[tex]\[r_m \circ R_{B^{\prime}, 270^{\circ}}\][/tex]
Hence, the correct answer is option 3.
