Question

regular hexagon abcdef is inscribed in a circle with center h.
a. what is the image of segment bc after a 180° clockwise rotation about point h?
type your answer in the box.
segment

b. what is the image of segment bc after a reflection over line fc?
type your answer in the box.
segment

Explanation:

Part a

• Step1: Understand the property of \(180^{\circ}\) rotation

A \(180^{\circ}\) rotation about the center \(H\) of a regular hexagon (which is inscribed in a circle) maps each point \(P\) to a point \(P'\) such that \(H\) is the mid - point of the segment \(PP'\).
In a regular hexagon \(ABCDEF\) inscribed in a circle with center \(H\), if we consider the vertices, for a \(180^{\circ}\) rotation about \(H\), vertex \(B\) is mapped to vertex \(E\) and vertex \(C\) is mapped to vertex \(D\).

• Step2: Determine the image of segment \(BC\)

Since a rotation is a rigid transformation (it preserves the length and the nature of the segment), the image of segment \(BC\) after a \(180^{\circ}\) clock - wise rotation about \(H\) is segment \(ED\).

Part b

• Step1: Understand the property of reflection

A reflection over a line \(l\) (in this case, line \(FC\)) is a transformation that maps each point \(P\) to a point \(P'\) such that the line \(l\) is the perpendicular bisector of the segment \(PP'\).
In a regular hexagon \(ABCDEF\) inscribed in a circle, line \(FC\) is a line of symmetry.

• Step2: Determine the image of segment \(BC\)

Vertex \(B\) is reflected over line \(FC\) to vertex \(CD\) (since the hexagon is symmetric about the line \(FC\)). The length of \(BC\) is equal to the length of \(CD\) (because it is a regular hexagon), and the reflection over line \(FC\) (a line of symmetry) maps segment \(BC\) to segment \(CD\).

Answer:

a. \(ED\)
b. \(CD\)