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 $EF$.

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.