Question

which rule describes the composition of transformations that maps △abc to △abc? options: ( r_{b, 270^circ} circ r_m ), ( r_{b, 90^circ} circ r_m ), ( r_m circ r_{b, 90^circ} ), ( r_m circ r_{b, 270^circ} ) (diagram shows triangles abc, abc, abc with horizontal line ( m ))

Explanation:

To solve the problem of identifying the composition of transformations mapping \(\triangle ABC\) to \(\triangle A''B''C''\), we analyze the transformations (reflection \(r_m\), rotation \(R_{B', \theta}\)) and their order (composition notation \(f \circ g\) means \(g\) first, then \(f\)):

Step 1: Recall Transformation Notation

• \(r_m\): Reflection over line \(m\).
• \(R_{B', \theta}\): Rotation about point \(B'\) by angle \(\theta\) (e.g., \(90^\circ\), \(270^\circ\)).
• Composition \(f \circ g\): Apply \(g\) first, then \(f\).

Step 2: Analyze the Image Sequence

1. First Transformation: The original \(\triangle ABC\) is first reflected over line \(m\) ( \(r_m\) ) to get an intermediate triangle (e.g., \(\triangle A'B'C'\)).
2. Second Transformation: Then, this intermediate triangle is rotated about \(B'\) by \(90^\circ\) ( \(R_{B', 90^\circ}\) ) to reach \(\triangle A''B''C''\).

Step 3: Match to Composition Notation

The composition is "rotate after reflecting," so the order is \(R_{B', 90^\circ} \circ r_m\) (apply \(r_m\) first, then \(R_{B', 90^\circ}\)).

1. Identify Transformations: The mapping involves a reflection (\(r_m\)) and a rotation (\(R_{B', 90^\circ}\)) about \(B'\).
2. Order of Transformations: Composition \( f \circ g \) means \( g \) (reflection \(r_m\)) is applied first, then \( f \) (rotation \(R_{B', 90^\circ}\)).
3. Match to Option: The option \( R_{B', 90^\circ} \circ r_m \) follows this order (reflect first, then rotate \(90^\circ\) about \(B'\)).

Answer:

\( R_{B', 90^\circ} \circ r_m \) (the option labeled \( R_{B', 90^\circ} \circ r_m \))