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Let's break down and solve the two parts of the problem related to the regular hexagon \( ABCDEF \) inscribed in a circle with center \( H \).

### Part (a): 120-degree clockwise rotation about point \( H \)
In a regular hexagon, a rotation of 120 degrees clockwise corresponds to moving each vertex forward by 2 positions, since each internal angle of the hexagon is \( 360^\circ / 6 = 60^\circ \). 

- **Starting segment**: \( BC \)
- After a **120-degree clockwise rotation**, each point moves two places forward:
  - \( B \) moves to \( D \)
  - \( C \) moves to \( E \)

So, the image of segment \( BC \) after the 120-degree clockwise rotation is **segment \( DE \)**.

### Part (b): Reflection over line \( FC \)
To find the image of segment \( BC \) after reflecting it over line \( FC \), we need to consider the symmetry of the hexagon. The reflection of a segment over a line in a regular hexagon swaps vertices symmetrically with respect to the line.

- Reflecting segment \( BC \) over line \( FC \) will swap vertices:
  - \( B \) reflects to \( E \) (since they are symmetric about line \( FC \))
  - \( C \) reflects to \( F \)

Thus, the image of segment \( BC \) after reflection over line \( FC \) is **segment \( EF \)**.

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Would you like further details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:
1. What happens to segment \( AB \) after a 60-degree clockwise rotation?
2. How does a reflection over line \( AB \) affect segment \( CD \)?
3. What is the image of vertex \( C \) after a 240-degree counterclockwise rotation?
4. How would the rotation angle change if we rotate 180 degrees instead of 120 degrees?
5. Can you find the image of segment \( DE \) after a reflection over line \( DA \)?

**Tip**: For regular polygons, rotations and reflections often follow cyclic patterns due to their symmetry, making it easier to predict outcomes.

4. Regular hexagon ABCDEF is inscribed in a circle with center H.
a. What is the image of segment BC after a 120-degree clockwise rotation about point H?
b. What is the image of segment BC after a reflection over line FC?

Explore transformations of a regular hexagon inscribed in a circle. This problem involves finding the image of segment BC after a 120-degree clockwise rotation about point H and a reflection over line FC. Suitable for students in Grades 8-10.

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