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To find the correct sequence of rigid motions that will take triangle \( XYZ \) onto triangle \( BCA \), let's analyze the transformations step-by-step. We have a few choices with translations, rotations, and reflections.

### Steps to Solve:
1. **Translation:** The first step across all options is to translate \( XYZ \) so that point \( Y \) (from triangle \( XYZ \)) aligns with point \( C \) (from triangle \( BCA \)). This translation will move the two triangles closer together, but not yet fully match their orientations.

2. **Rotation:** The next step is a rotation around point \( C \) (which is now coinciding with \( Y' \) after the translation). The rotation needs to align either \( X \) with \( B \) or \( A \), depending on the specific answer choice.

3. **Reflection:** The final step involves a reflection to fully match triangle \( XYZ \) onto \( BCA \). The correct line of reflection (either \( CB \) or \( AC \)) will depend on which side the triangle lands after the rotation.

### Analyzing Options:
- **Option A:** Translates \( XYZ \) using line \( YC \), rotates using \( C \) so \( X' \) coincides with \( B \), then reflects across line \( CB \).
- **Option B:** Similar to Option A but reflects across line \( AC \) instead.
- **Option C:** Translates \( XYZ \), rotates so \( X' \) coincides with \( A \), and reflects across \( CB \).
- **Option D:** Translates \( XYZ \), rotates so \( X' \) coincides with \( A \), and reflects across \( AC \).

### Solution
The correct answer is **Option A**. This sequence (Translate along \( YC \), Rotate so \( X' \) coincides with \( B \), Reflect across \( CB \)) should map \( XYZ \) onto \( BCA \) correctly.

Let me know if you'd like further details or have any questions. 

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Here are 5 follow-up questions:
1. How does a translation affect the orientation of a shape?
2. What properties of triangles remain unchanged under rigid transformations?
3. How do you determine the center and angle for a rotation?
4. Why is reflection necessary after a translation and rotation in this case?
5. Can all congruent triangles be mapped onto each other using only rigid transformations?

**Tip:** When determining transformations, visualizing each step on a diagram helps clarify how shapes move and align.

The triangles are congruent. Which sequence of rigid motions will take triangle XYZ onto triangle BCA?

Explore how to map triangle XYZ onto triangle BCA using a sequence of rigid motions, including translation, rotation, and reflection. This problem covers congruence, triangle transformations, and is suitable for students in Grades 8-10.

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