https://file.aiquickdraw.com/mathbot/file/1726752466983ai40w06u.jpg

To map Figure B onto Figure C, we need to determine the transformations required, which can involve translations, rotations, reflections, and dilations. Let’s analyze the position and orientation of the two figures.

### Observations:
1. **Figure B** is in quadrant I, and **Figure C** is in quadrant IV, rotated and reflected compared to Figure B.
2. The shapes appear identical in size, so no dilation (scaling) is needed.
3. Figure B has been rotated and possibly reflected, followed by a translation.

### Steps to map Figure B to Figure C:

1. **Rotation**: 
   - We can rotate Figure B by 180° around the origin to get a shape that has a similar orientation to Figure C. This moves Figure B to the fourth quadrant, making it upside down compared to its original position.

2. **Translation**: 
   - After rotating, the figure will be positioned near (-x, -y) coordinates relative to Figure B’s original position. To align it perfectly with Figure C, we need to translate the figure:
     - Translate the rotated image by a vector of approximately \((\Delta x, \Delta y) = (3, -7)\), which moves the figure to match Figure C.

### Summary of Transformations:
1. **Rotate** Figure B 180° around the origin.
2. **Translate** the rotated figure by the vector \((3, -7)\) to map it onto Figure C.

Would you like more details on any of the transformations?

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### Related Questions:
1. What happens if we apply a reflection instead of rotation to Figure B?
2. How would the transformation change if Figure B were dilated?
3. Can you describe a reflection that could map Figure B onto Figure C directly?
4. How can we check if these transformations preserve the shape and size of the figure?
5. What is the effect of rotating a figure 90° instead of 180°?

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**Tip**: When working with transformations, always perform rotations and reflections first, followed by translations to avoid unnecessary complexity.

Determine a series of transformations that would map Figure B onto Figure C.

Learn how to map Figure B onto Figure C using geometric transformations, including a 180° rotation and translation. This problem covers key concepts in geometric transformations and is suitable for students in Grades 8-10.

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