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.

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.