Question

how can a translation and a rotation be used to map \\( \triangle h j k \\) to \\( \triangle l m n \\)?
translate \\( h \\) to \\( n \\)
and rotate about \\( h \\)
until \\( h k \\) lies on the
line containing \\( l n \\).
translate \\( h \\) to \\( l \\)
and rotate about \\( h \\)
until \\( h k \\) lies on the
line containing \\( l m \\).
translate \\( k \\) to \\( m \\)
and rotate about \\( k \\)
until \\( h k \\) lies on the
line containing \\( l m \\).
translate \\( k \\) to \\( n \\)
and rotate about \\( k \\)
until \\( h k \\) lies on the
line containing \\( l n \\).

Explanation:

Step1: Analyze translation

Translation moves a figure without rotation or reflection. We want to map \(\triangle HJK\) to \(\triangle LMN\). If we translate \(K\) to \(M\), we start to align the two triangles.

Step2: Analyze rotation after translation

After translating \(K\) to \(M\), we then rotate about \(K\) (now at \(M\) after translation). Rotating until \(HK\) (after translation) lies on the line containing \(LM\) will help map \(\triangle HJK\) to \(\triangle LMN\) as the sides and angles will match up due to the properties of translation (preserves shape and size) and rotation (preserves shape and size).

Answer:

Translate \(K\) to \(M\) and rotate about \(K\) until \(HK\) lies on the line containing \(LM\).