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To determine the correct sequence of rigid motions that maps triangle \( DEF \) onto triangle \( BAC \), let's analyze the choices provided:

1. **Step 1: Translation**  
   - All options suggest translating \( DEF \) so that point \( E \) matches point \( A \). This aligns the triangles partially, setting up for the next transformation.

2. **Step 2: Rotation**  
   - After translating, the next action involves rotating around a specified point. We need to see which rotation aligns another point correctly:
     - Some options propose rotation to match \( D' \) with \( B \), while others suggest matching \( D' \) with \( C \).
     - Observing the layout, rotating around \( A \) to bring \( D' \) to \( B \) seems feasible, as it keeps the configuration similar to \( BAC \).

3. **Step 3: Reflection**  
   - Finally, we consider reflection:
     - To fully align the triangles, reflecting across the line that includes two of the target triangle's vertices should complete the transformation.
     - Reflecting across \( AB \) (the line containing points \( A \) and \( B \)) would match all vertices.

Based on these steps, the correct answer appears to be the first option:

- **Translate \( DEF \) from \( E \) to \( A \). Rotate \( D'E'F' \) using \( A \) as the center so that \( D' \) coincides with \( B \). Reflect \( D''E''F'' \) across line \( AB \).**

Would you like more details on the transformations or a breakdown of each rigid motion?

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Here are five related questions to expand on this:

1. How do you determine the center of rotation in transformations?
2. What is the difference between reflection and rotation in rigid motions?
3. How would the transformations change if we needed to map \( DEF \) onto a triangle with different vertex labels?
4. Can a sequence of transformations ever produce different final results if performed in a different order?
5. Why is it important that the triangles are congruent when using rigid motions?

**Tip:** When using rigid motions, focus on matching corresponding points systematically to ensure accuracy in transformations.

The triangles are congruent. Which sequence of rigid motions takes triangle DEF onto triangle BAC?

Learn how to map triangle DEF onto triangle BAC using a sequence of rigid transformations, including translation, rotation, and reflection. This problem covers key concepts in congruence and transformations, suitable for students in Grades 8-10.

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