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The problem is asking for the set of transformations that transform triangle ABC into triangle A'B'C'. We can solve this by analyzing the transformation step by step.

1. **Rotation**:
   Looking at the orientation of triangle ABC and triangle A'B'C', it appears that the triangle is rotated **180 degrees counterclockwise** about the origin. After a 180° rotation about the origin, the coordinates of any point \((x, y)\) become \((-x, -y)\). Checking the coordinates, we see that this transformation holds.

   For example:
   - \( A(1, 4) \) maps to \( A'(-1, -4) \),
   - \( B(2, 2) \) maps to \( B'(-2, -2) \),
   - \( C(4, 3) \) maps to \( C'(-4, -3) \).

2. **Translation**:
   After the rotation, triangle A'B'C' appears to have been translated **5 units to the right**. This means we add 5 to the x-coordinates of each point.

   For example:
   - \( A'(-1, -4) \) becomes \( A''(4, -4) \),
   - \( B'(-2, -2) \) becomes \( B''(3, -2) \),
   - \( C'(-4, -3) \) becomes \( C''(1, -3) \).

These transformations match the given triangle A'B'C'. Therefore, the correct transformation is:

**A 180 degree counterclockwise rotation about the origin followed by a translation 5 units to the right**, which corresponds to the answer you selected.

Would you like further details or clarifications on this?

Here are some related questions to think about:
1. How do you calculate the coordinates of a point after a 90° rotation around the origin?
2. What would happen if you translated the triangle before rotating it?
3. How does a reflection transformation differ from a rotation?
4. How can you determine the center of rotation for a given transformation?
5. What are the effects of a negative translation versus a positive translation?

**Tip**: When applying multiple transformations, always consider the order, as the result can differ depending on whether you rotate first or translate first.

What set of transformations is performed on triangle ABC to form triangle A'B'C'?

Discover how to transform triangle ABC into triangle A'B'C' using a 180-degree rotation followed by a translation of 5 units to the right. This detailed solution covers the steps involved in combining geometric transformations, suitable for students in Grades 8-10.

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