Given segment $BC$ in the plane, what is the image of segment $BC$ after a $120$-degree clockwise rotation about point $H$?

• A. Segment $B^{\text{'}\text{'}}{C}^{\text{'}\text{'}}$
• B. Segment $B^{\text{'}}{C}^{\text{'}}$
• C. Segment $HB$
• D. Segment $B{C}^{\text{'}}$

Answer: (B)

Verified step by step guidance

1. Identify the center of rotation, which is point \(H\), and note the angle of rotation is \$120$ degrees clockwise.
2. Recall that a rotation of \(\theta\) degrees clockwise about a point \(H\) can be represented by rotating each point of the segment around \(H\) by \(\theta\) degrees in the clockwise direction.
3. To find the image of segment \(BC\), rotate both endpoints \(B\) and \(C\) around \(H\) by \$120\( degrees clockwise. This involves calculating the new coordinates of \)B'\( and \)C'$ after rotation.
4. Use the rotation formulas for a point \((x,y)\) rotated about \((h,k)\) by \(\theta\) degrees clockwise: \( x' = h + (x - h) \cos(\theta) + (y - k) \sin(\theta) \) \( y' = k - (x - h) \sin(\theta) + (y - k) \cos(\theta) \) where \(\theta = 120^\circ\).
5. After finding the coordinates of \(B'\) and \(C'\), connect these points to form the image segment \(B'C'\), which is the rotated image of segment \(BC\) about point \(H\).