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.