Question

two rigid transformations are used to map \\(\delta hjk\\) to \\(\delta lmn\\). the first is a translation of vertex h to vertex l. what is the second transformation?

a reflection across the line containing \\(\overline{hk}\\)
a rotation about point h
a reflection across the line containing \\(\overline{hj}\\)
a rotation about point k

Explanation:

Analyze the mapping of vertices

Using the Congruence Mapping knowledge point

• The problem states that two rigid transformations map \(\Delta HJK\) to \(\Delta LMN\).
• This implies the vertex correspondences are:
• \(H \to L\)
• \(J \to M\)
• \(K \to N\)

Analyze the first transformation

Using the Rigid Transformations knowledge point

• The first transformation is a translation mapping vertex \(H\) to vertex \(L\).
• Let \(\Delta H'J'K'\) be the image of \(\Delta HJK\) after this translation.
• Since \(H\) is translated to \(L\), we have \(H' = L\).
• The segment \(H'K'\) is now parallel to \(HK\), starting at \(L\).

Determine the second transformation

Using the Rigid Transformations knowledge point

• After the translation, the image \(\Delta H'J'K'\) shares vertex \(H' = L\) with \(\Delta LMN\).
• To complete the mapping to \(\Delta LMN\), we must rotate the triangle about the shared vertex.
• Since the shared vertex is \(H' = L\) (which corresponds to the original point \(H\)), the rotation must be centered at this point.
• Therefore, the second transformation is a rotation about point \(H\).

Answer:

• a reflection across the line containing \(\overline{HK}\)
• a rotation about point \(H\) (Correct answer)
• a reflection across the line containing \(\overline{HJ}\)
• a rotation about point \(K\)