Question

which rigid transformation would map △mzk to △qzk?
○ a rotation about point k
○ a reflection across the line containing \\(\overline{mz}\\)
○ a reflection across the line containing \\(\overline{zk}\\)
○ a rotation about point z

Explanation:

To determine the rigid transformation mapping \( \triangle MZK \) to \( \triangle QZK \), we analyze the diagram:

• \( ZK \) is perpendicular to \( MQ \) and bisects \( MQ \) (since \( Z \) is the midpoint, \( MZ = ZQ \), and \( \angle KZM=\angle KZQ = 90^\circ \)).
• A reflection across the line containing \( \overline{ZK} \) would map \( M \) to \( Q \) (as \( ZK \) is the perpendicular bisector of \( MQ \)) and keep \( Z \) and \( K \) fixed. This matches the congruence of \( \triangle MZK \) and \( \triangle QZK \).
• Rotation about \( K \) or \( Z \) would not align the triangles as neatly, and reflection across \( \overline{MZ} \) would not map \( M \) to \( Q \).

Answer:

a reflection across the line containing \( \overline{ZK} \)