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 ?. Type the answer in the box below.Segment: b. What is the image of segment BC after a reflection over line FC? Type the answer in the box below. Segment:

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!

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 the geometric transformations of segment BC in a regular hexagon. This problem involves a 120-degree clockwise rotation about the center and a reflection over a line. Learn how to apply rotation and reflection formulas, and understand the symmetry of regular polygons. Suitable for students in Grades 9-12.

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