Question

which rule describes the composition of transformations that maps $\triangle abc$ to $\triangle abc$?
\\( r_{b, 90^{\circ}} \circ t_m \\)\\( t_m \circ r_{b, 90^{\circ}} \\)\\( t_m \circ r_{b, 270^{\circ}} \\)\\( r_{b, 270^{\circ}} \circ t_m \\)

Explanation:

Step1: Analyze Transformation Order

First, identify the transformations. The notation \( T_m \) is a translation, \( R_{B', \theta} \) is a rotation about \( B' \) by \( \theta \) degrees. The composition order matters: \( T_m \circ R_{B', 90^\circ} \) means rotate first then translate? Wait, no—composition is rightmost first. Wait, actually, \( f \circ g \) means \( g \) then \( f \). So for \( T_m \circ R_{B', 90^\circ} \), it's rotate \( 90^\circ \) about \( B' \) first, then translate by \( T_m \)? Wait, no, let's check the diagram. The original triangle \( \triangle ABC \), first maybe rotate about \( B \) (or \( B' \))? Wait, the target is \( \triangle A''B'C'' \). Let's see the steps: first, translate \( \triangle ABC \) to \( \triangle A'C'B' \) (maybe \( T_m \)), then rotate \( 90^\circ \) about \( B' \)? Wait, no, the composition \( T_m \circ R_{B', 90^\circ} \) would be rotate \( 90^\circ \) about \( B' \) first, then translate. Wait, maybe the correct order is: first rotate \( 90^\circ \) about \( B' \), then translate? No, wait the options: let's check the labels. The first transformation is rotation about \( B' \) by \( 90^\circ \), then translation? Wait, no, the composition \( T_m \circ R_{B', 90^\circ} \) is translate after rotating. Wait, maybe the correct answer is \( T_m \circ R_{B', 90^\circ} \)? Wait, no, let's re-examine. The original triangle \( ABC \), to get to \( A''B'C'' \), first we rotate \( 90^\circ \) about \( B' \) (or a point), then translate? Wait, the options: the first option is \( T_m \circ R_{B', 90^\circ} \), which is translate after rotating \( 90^\circ \) about \( B' \). Wait, maybe the correct order is rotate \( 90^\circ \) about \( B' \) first, then translate by \( T_m \), which is \( T_m \circ R_{B', 90^\circ} \) (since composition is right function first, then left). So the correct rule is \( T_m \circ R_{B', 90^\circ} \)? Wait, no, the second option is \( R_{B', 90^\circ} \circ T_m \), which is translate first, then rotate. Wait, let's look at the diagram. The original \( \triangle ABC \), first translate to \( \triangle A'C'B' \) (so \( T_m \)), then rotate \( 90^\circ \) about \( B' \) to get \( \triangle A''B'C'' \). So the composition is rotate after translate, which is \( R_{B', 90^\circ} \circ T_m \)? Wait, no—composition \( f \circ g \) is \( g \) then \( f \). So if we translate first (\( T_m \)), then rotate (\( R_{B', 90^\circ} \)), the composition is \( R_{B', 90^\circ} \circ T_m \). Wait, but the first option is \( T_m \circ R_{B', 90^\circ} \) (rotate first, then translate), the second is \( R_{B', 90^\circ} \circ T_m \) (translate first, then rotate). Let's check the diagram: the middle triangle is \( \triangle A'C'B' \), which is a translation of \( \triangle ABC \) (so \( T_m \) takes \( ABC \) to \( A'C'B' \)), then rotating \( 90^\circ \) about \( B' \) takes \( A'C'B' \) to \( A''B'C'' \). So the composition is rotate after translate, which is \( R_{B', 90^\circ} \circ T_m \)? Wait, no, the option is \( T_m \circ R_{B', 90^\circ} \)? Wait, maybe I got the composition order wrong. Wait, in function composition, \( (f \circ g)(x) = f(g(x)) \), so \( g \) is applied first, then \( f \). So if we have \( T_m \circ R_{B', 90^\circ} \), that's \( R_{B', 90^\circ} \) first, then \( T_m \). If we have \( R_{B', 90^\circ} \circ T_m \), that's \( T_m \) first, then \( R_{B', 90^\circ} \). Looking at the diagram, \( \triangle ABC \) is first translated (via \( T_m \)) to \( \triangle A'C'B' \), then rotated \( 90^\circ \) about \( B' \) to \( \triangle A''B'C'' \). So the composition is \( R_{B', 90^\circ} \circ T_m \)? Wait, no, the second option is \( R_{B', 90^\circ} \circ T_m \), which is translate first, then rotate. But let's check the labels. Wait, maybe the rotation is \( 90^\circ \) clockwise or counter? Wait, the diagram: \( \triangle A'C'B' \) to \( \triangle A''B'C'' \) is a \( 90^\circ \) rotation about \( B' \). So first, translate \( ABC \) to \( A'C'B' \) ( \( T_m \) ), then rotate \( 90^\circ \) about \( B' \) ( \( R_{B', 90^\circ} \) ), so the composition is \( R_{B', 90^\circ} \circ T_m \), which is the second option? Wait, no, the first option is \( T_m \circ R_{B', 90^\circ} \), second is \( R_{B', 90^\circ} \circ T_m \), third is \( R_{B', 180^\circ} \circ T_m \), fourth is \( T_m \circ R_{B', 180^\circ} \). Wait, the angle: is it \( 90^\circ \) or \( 180^\circ \)? The triangle \( A'C'B' \) to \( A''B'C'' \): the rotation from \( A'C'B' \) (right triangle) to \( A''B'C'' \) (left triangle) is \( 90^\circ \) rotation (since the legs are perpendicular). So \( 90^\circ \), not \( 180^\circ \). So between the first two options: \( T_m \circ R_{B', 90^\circ} \) (rotate then translate) or \( R_{B', 90^\circ} \circ T_m \) (translate then rotate). Wait, the original \( ABC \): if we rotate first about \( B' \), where is \( B' \) relative to \( B \)? \( B' \) is the image of \( B \) after translation? Wait, maybe \( B \) translates to \( B' \) (so \( T_m \) moves \( B \) to \( B' \)), then rotate \( 90^\circ \) about \( B' \). So the order is: translate \( ABC \) to \( A'C'B' \) ( \( T_m \) ), then rotate \( 90^\circ \) about \( B' \) to \( A''B'C'' \). So the composition is \( R_{B', 90^\circ} \circ T_m \), which is the second option? Wait, no, the first option is \( T_m \circ R_{B', 90^\circ} \), which would be rotate \( ABC \) about \( B' \) (but \( B' \) is not the same as \( B \) before translation), so that doesn't make sense. So the correct order is translate first ( \( T_m \) ), then rotate \( 90^\circ \) about \( B' \), so the composition is \( R_{B', 90^\circ} \circ T_m \), which is the second option: \( R_{B', 90^\circ} \circ T_m \). Wait, but let's check the option labels. The second option is \( R_{B', 90^\circ} \circ T_m \), so that's the correct composition: translate \( T_m \) first, then rotate \( 90^\circ \) about \( B' \).

Step2: Confirm with Diagram

Looking at the diagram, \( \triangle ABC \) (top middle) translates to \( \triangle A'C'B' \) (top right) via \( T_m \) (the horizontal arrow), then rotates \( 90^\circ \) about \( B' \) to \( \triangle A''B'C'' \) (top left). So the composition is rotate after translate, which is \( R_{B', 90^\circ} \circ T_m \), matching the second option.

Answer:

\( R_{B', 90^\circ} \circ T_m \) (the second option, assuming the options are labeled as:
A. \( T_m \circ R_{B', 90^\circ} \)
B. \( R_{B', 90^\circ} \circ T_m \)
C. \( R_{B', 180^\circ} \circ T_m \)
D. \( T_m \circ R_{B', 180^\circ} \)

Then the answer is B. \( R_{B', 90^\circ} \circ T_m \))