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To determine the correct rule(s) for translating \( \triangle RST \) to \( \triangle R'S'T' \), let's analyze the movement of each vertex of the triangle.

1. **Identify the Translation Vector:**
   Let's look at how point \( R \) moves to \( R' \), \( S \) moves to \( S' \), and \( T \) moves to \( T' \):
   - **\( R \) to \( R' \)**: The point \( R \) is at approximately \((-3, 4)\), and \( R' \) is at approximately \((4, 1)\).
     - To go from \((-3, 4)\) to \((4, 1)\), we move **7 units to the right** (increasing \( x \) by 7) and **3 units down** (decreasing \( y \) by 3).

   This pattern holds for all points of the triangle, confirming that the translation vector is \((+7, -3)\).

2. **Match with Given Options:**
   - **Option C**: "7 units right; 3 units down" — This matches our calculation.
   - **Option D**: \( T_{(7, -3)} (x, y) = (x + 7, y - 3) \) — This also matches, as it represents a translation by \((+7, -3)\).

Therefore, the correct answers are:
- **C. 7 units right; 3 units down**
- **D. \( T_{(7, -3)} (x, y) = (x + 7, y - 3) \)**

Would you like further details on how to approach translation problems, or have any questions?

### Related Questions:
1. How do we verify the translation vector when given only two points?
2. What is the effect of a negative translation vector on the direction of movement?
3. Can multiple translations yield the same result for different shapes?
4. How would you graphically determine the translation vector without coordinates?
5. How does translation differ from reflection in terms of rules?

### Tip:
For any translation, check a few key points to confirm the movement, ensuring consistency across all vertices.

Select all the rules for the translation of triangle RST.

Learn how to determine the rules for the translation of triangle RST to triangle R'S'T' in a coordinate plane. This problem covers essential geometry concepts, including translation rules and coordinate transformations, making it suitable for Grade 8 students.

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