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The problem asks for three rotations of the line segment \( CD \) by \( 180^\circ \) about different points: \( D \), \( E \), and \( M \). Here’s the solution for each part:

### a. Rotate segment \( CD \) \( 180^\circ \) around point \( D \)
- When you rotate a line segment \( 180^\circ \) around one of its endpoints (in this case \( D \)), the segment flips to the opposite side. Since \( D \) remains fixed, the point \( C \) will move to the other side of \( D \) such that the new position of \( C \) (let's call it \( C' \)) lies on the same straight line but on the opposite side at an equal distance from \( D \). The direction is reversed, so \( C' \) will now be to the right of \( D \).

### b. Rotate segment \( CD \) \( 180^\circ \) around point \( E \)
- For this rotation, the line segment \( CD \) is rotated around the point \( E \), which is located off the line. A \( 180^\circ \) rotation means every point on the line moves to the opposite side of \( E \). \( D \) will move to a point \( D' \) directly opposite its current position relative to \( E \), and \( C \) will move to a new position \( C' \). The entire segment will maintain the same length but will be flipped, appearing on the opposite side of \( E \).

### c. Rotate segment \( CD \) \( 180^\circ \) around point \( M \)
- When you rotate around the midpoint \( M \), both points \( C \) and \( D \) will exchange places. Since \( M \) is the midpoint, a \( 180^\circ \) rotation will swap \( C \) and \( D \), so \( C \) moves to \( D \)’s position, and \( D \) moves to \( C \)’s position, creating a symmetric transformation.

Would you like further details on any of these steps? 

Here are five questions that expand on this topic:
1. How does the distance of the rotation point from a line segment affect the result of the rotation?
2. What would happen if you rotated segment \( CD \) by \( 90^\circ \) instead of \( 180^\circ \)?
3. How can you geometrically prove that rotating around a midpoint swaps the endpoints of a line segment?
4. What properties of the figure remain unchanged after a \( 180^\circ \) rotation?
5. How would these rotations change if you used a different rotation center that is not on the plane of the line?

**Tip:** Rotating a figure by \( 180^\circ \) always results in a figure that looks identical but reversed in direction or orientation.

Rotate segment CD 180° around points D, E, and M.

This problem involves rotating a line segment CD 180° around three distinct points—D, E, and M. Explore how geometric transformations affect line segments in a coordinate plane. This is suitable for students in Grades 6-9 studying rotations and transformations.

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