Let's break down each part of the question step by step.

Part a: 120-degree clockwise rotation of segment BC about point H

Since $\triangle ABCDEF$ is a regular hexagon inscribed in a circle, all its central angles (angles subtended by the sides at the center) are equal to $60^\circ$. When you rotate any point of the hexagon by $120^\circ$ clockwise about its center $H$, each vertex moves two positions clockwise along the perimeter of the hexagon.

• Segment $BC$ connects vertices $B$ and $C$.
• A $120^\circ$ clockwise rotation will move:
  • Point $B$ to point $D$,
  • Point $C$ to point $E$.

Thus, after the rotation, segment $BC$ becomes segment DE.

Answer to part a: Segment DE

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Part b: Reflection of segment BC over line FC

Next, let's reflect segment $BC$ over line $FC$. In a regular hexagon, reflection over a line through two vertices will map other vertices symmetrically.

• Reflecting segment $BC$ over line $FC$ swaps:
  • Point $B$ with point $E$,
  • Point $C$ with itself (since $C$ lies on the line of reflection, it stays fixed).

Thus, after the reflection, segment $BC$ becomes segment EC.

Answer to part b: Segment EC

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If you'd like any more details or have additional questions, feel free to ask!

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Related Questions:

1. What is the image of segment $AB$ after a $60^\circ$ clockwise rotation about point $H$?
2. How does a $180^\circ$ rotation about the center affect the orientation of the hexagon?
3. What is the image of segment $EF$ after a reflection over line $AC$?
4. If we reflect the entire hexagon over the line $FC$, what happens to the shape?
5. How would a reflection over line $AB$ affect the position of point $C$?

Tip: For regular polygons, rotations and reflections often result in symmetry-based transformations!