https://file.aiquickdraw.com/mathbot/file/17295488918262gw9q146.jpg

The image shows two quadrilaterals, \( OPQR \) and \( O'P'Q'R' \), along with a question asking to identify the sequence of transformations that maps \( OPQR \) onto \( O'P'Q'R' \). The transformations to choose from involve translations, rotations, and reflections.

### To analyze the transformation:
1. **Position of \( OPQR \) and \( O'P'Q'R' \):**
   - \( OPQR \) is in the second quadrant (coordinates are negative for \( x \)-values).
   - \( O'P'Q'R' \) is in the first quadrant (positive \( x \)-values).

2. **Possible transformations:**
   - **Translation**: There is a clear shift horizontally and vertically from \( OPQR \) to \( O'P'Q'R' \).
   - **Reflection**: Since the shapes are mirrored across either axis (most likely across the \( y \)-axis).
   - **Rotation**: It's also possible that a rotation was involved based on their relative positioning.

### Examining each choice:
- **Option 1**: "A translation left 12 units followed by a reflection across the \( x \)-axis."  
  This doesn't seem correct because the shapes are not mirrored across the \( x \)-axis.

- **Option 2**: "A rotation 90° clockwise around the origin followed by a translation up 12 units."  
  This option seems plausible because rotating \( OPQR \) by 90° clockwise would align its orientation with \( O'P'Q'R' \), and then a translation upward could place it correctly.

- **Option 3**: "A translation left 2 units and up 9 units followed by a reflection across the \( y \)-axis."  
  This doesn't seem accurate, as the translation described here doesn't align the two shapes perfectly.

### Correct Answer: 
**Option 2**: A rotation of 90° clockwise around the origin, followed by a translation up 12 units, maps \( OPQR \) onto \( O'P'Q'R' \).

Would you like further details on this transformation process, or do you have any specific questions?

Here are 5 related questions to explore this topic further:
1. How do you determine the center of rotation for a geometric transformation?
2. What are the rules for reflections across the \( x \)- and \( y \)-axes?
3. How do you identify if a shape has undergone a translation only?
4. What is the difference between a 90° clockwise and counterclockwise rotation?
5. How can you calculate the distance between corresponding points after a translation?

**Tip**: When dealing with transformations, always check if a reflection or rotation is involved by comparing the orientation and relative position of corresponding points.

Which sequence of transformations maps OPQR onto O'P'Q'R'?

Learn how to map quadrilateral OPQR onto O'P'Q'R' using transformations. This problem explores translations, rotations, and reflections in the coordinate plane, suitable for students in Grades 8-10.

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