https://file.aiquickdraw.com/mathbot/file/1727815989134utfykbac.jpg

The question asks for transformations (translations, rotations, and reflections) that would take Polygon P (the green one) to Polygon Q (the blue one). Let's go through each of the options to determine if they are correct.

1. **Translate so that A is taken to J. Then rotate 120° clockwise around point D.**
   - This involves moving Polygon P so that point A matches point J, followed by a 120° clockwise rotation around point D.
   - This transformation seems plausible, as the rotation would align the two polygons. **This is a valid transformation.**

2. **Reflect over DX.**
   - A reflection over the line DX would flip the polygon across the line DX. However, this would not map Polygon P onto Polygon Q. **This is not a valid transformation.**

3. **Rotate 180° counterclockwise around D and reflect over AD.**
   - A 180° counterclockwise rotation around D would position the polygon upside down, and a reflection over line AD would then flip it. This seems to correctly map Polygon P onto Q. **This is a valid transformation.**

4. **Rotate 120° clockwise around point D.**
   - A 120° clockwise rotation around point D without any translation seems to work in aligning Polygon P with Q. **This is a valid transformation.**

5. **Reflect over CD and rotate 60° clockwise around point D.**
   - A reflection over line CD followed by a 60° clockwise rotation around D does not align Polygon P with Q. **This is not a valid transformation.**

### Conclusion:
The correct transformations are:
- **Translate so that A is taken to J. Then rotate 120° clockwise around point D.**
- **Rotate 180° counterclockwise around D and reflect over AD.**
- **Rotate 120° clockwise around point D.**

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the mathematical definition of a reflection?
2. How do you calculate the result of a 120° rotation in a coordinate plane?
3. What is the process of combining transformations like translation and rotation?
4. How does a reflection over a line affect the orientation of a shape?
5. Can any sequence of transformations map any polygon to another congruent polygon?

**Tip:** When performing a series of transformations, visualize each step independently to ensure the final position aligns with the target shape.

Select all sequences of translations, rotations, and reflections that would take Polygon P to Polygon Q.

This problem involves using translation, rotation, and reflection transformations to map Polygon P to Polygon Q. Explore step-by-step how 120° rotations, reflections over lines, and translations are applied to solve the problem, suitable for Grades 8-10.

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