Question

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

Explanation:

Step1: Analyze translation

Translation moves a figure without rotation or reflection. We need to map a vertex of \(\triangle HJK\) to a vertex of \(\triangle LMN\).

Step2: Analyze rotation

Rotation is a transformation about a point. After translation, we rotate to align the sides.
For \(\triangle HJK\) and \(\triangle LMN\), we should translate \(K\) to \(N\) (to get one - to - one vertex correspondence). Then rotate about \(K\) (since we translated \(K\) and rotation about the translated point's original position in the pre - image (which is equivalent in transformation terms) until \(\overline{HK}\) lies on the line containing \(\overline{LN}\)

Answer:

Translate \(K\) to \(N\) and rotate about \(K\) until \(\overline{HK}\) lies on the line containing \(\overline{LN}\). So the correct option is “Translate \(K\) to \(N\) and rotate about \(K\) until \(\overline{HK}\) lies on the line containing \(\overline{LN}\)”