Question

the triangles shown are congruent by the sss congruence theorem. the diagram shows the sequence of three rigid transformations used to map △abc onto △abc. what is the sequence of the transformations?
○ rotation, then reflection, then translation
○ rotation, then translation, then reflection
○ translation, then reflection, then rotation
○ translation, then rotation, then reflection

Explanation:

1. First, analyze the initial transformation from \( \triangle ABC \) to \( \triangle A'B'C' \): This looks like a rotation (changing the orientation around a point).
2. Then, from \( \triangle A'B'C' \) to the next step (before reflection), a translation (sliding the figure) would move it closer to the final position.
3. Finally, a reflection (flipping over a line) would map it to \( \triangle A''B''C'' \) as the orientation and position match after these steps. Checking the options, the sequence rotation, then translation, then reflection (option B) fits. Wait, no—wait, let's re - examine. Wait, actually, first, to get from \( \triangle ABC \) to a position, maybe first rotation, then translation, then reflection? Wait, no, let's think about rigid transformations. Rigid transformations preserve shape and size. Let's check the order: rotation (changes angle), then translation (moves), then reflection (flips). The correct sequence is rotation, then translation, then reflection? Wait, no, the option is "rotation, then translation, then reflection" (second option). Wait, let's confirm:

• Rotation: Changes the orientation around a center.
• Translation: Moves the figure without rotating or reflecting.
• Reflection: Flips over a line.

Looking at the diagram, the first transformation (from \( ABC \) to \( A'B'C' \)) is a rotation. Then, moving \( A'B'C' \) to a position closer (translation), then reflecting to get \( A''B''C'' \). So the sequence is rotation, then translation, then reflection.

Answer:

B. rotation, then translation, then reflection