Question

1. regular hexagon abcdef is inscribed in a circle with center h.

a. what is the image of segment bc after a 180° clock - wise rotation about point e?
b. what is the image of segment bc after a reflection over line fc?

Explanation:

Step1: Analyze 180 - degree rotation

A 180 - degree clock - wise rotation about a point maps a point \(P(x,y)\) to \(P'(-x,-y)\) relative to the center of rotation. In a regular hexagon \(ABCDEF\) inscribed in a circle with center \(H\), when we rotate segment \(BC\) 180 degrees clockwise about point \(E\), we consider the properties of a regular hexagon. The central angle of a regular hexagon is \(\frac{360^{\circ}}{6}=60^{\circ}\). After a 180 - degree rotation about \(E\), the image of segment \(BC\) is segment \(AF\).

Step2: Analyze reflection

When we reflect a segment \(BC\) over line \(FC\), we use the property of reflection. In a regular hexagon, line \(FC\) is a line of symmetry. The image of segment \(BC\) after reflection over line \(FC\) is segment \(CD\).

Answer:

a. Segment \(AF\)
b. Segment \(CD\)