https://file.aiquickdraw.com/mathbot/file/1727403518535u44fqwph.jpg

The task asks to rotate the pentagon \(ABCDE\) by 180° clockwise about point \(C\), and label the corresponding points \(A', B', D', E'\).

### Steps to Rotate the Pentagon 180°:
1. **Locate point C**: Since the rotation is centered around point \(C\), this point will remain fixed during the rotation.

2. **Determine new positions of other points**:
   - For a 180° rotation, each point will end up on the opposite side of \(C\), at the same distance from \(C\) as before but in the opposite direction.
   
3. **Mapping each vertex**:
   - **Point A**: Count the number of units \(A\) is from \(C\). It is 3 units right and 2 units up. After the 180° rotation, \(A'\) will be 3 units left and 2 units down from \(C\).
   - **Point B**: \(B\) is 2 units right and 2 units up from \(C\). After the rotation, \(B'\) will be 2 units left and 2 units down from \(C\).
   - **Point D**: \(D\) is 2 units left and 1 unit down from \(C\). After the rotation, \(D'\) will be 2 units right and 1 unit up from \(C\).
   - **Point E**: \(E\) is 1 unit left and 2 units down from \(C\). After the rotation, \(E'\) will be 1 unit right and 2 units up from \(C\).

### Final Positions:
- \(A' = (-3, -2)\)
- \(B' = (-2, -2)\)
- \(D' = (2, 1)\)
- \(E' = (1, 2)\)

Would you like me to provide more details on any step? 

Here are 5 follow-up questions to expand on this:
1. What is the general rule for rotating a point 180° around another point?
2. How can we apply matrix transformations for such rotations?
3. How do clockwise and counterclockwise rotations differ geometrically?
4. Can you explain the symmetry observed in a 180° rotation?
5. What other geometric properties remain unchanged during a rotation?

**Tip**: When rotating, always ensure to measure distances carefully from the center of rotation for accurate placement.

Draw the rotation of pentagon ABCDE clockwise 180° about point C. Label the corresponding points in the image with A', B', C', D', and E'.

This problem involves rotating a pentagon ABCDE by 180° clockwise about a fixed point C and identifying the new coordinates of the vertices A', B', D', and E'. Key geometry concepts such as rotation and coordinate transformations are applied. Suitable for students in Grades 6-8.

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