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?

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°?

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