Question

is \\( \triangle def \\) congruent to \\( \triangle ghi \\)? justify your answer using rigid motions.

yes. a translation and a reflection can map \\( \triangle def \\) onto \\( \triangle ghi \\).

yes. a translation and a rotation can map \\( \triangle def \\) onto \\( \triangle ghi \\).

no. it is not possible to map \\( \triangle def \\) onto \\( \triangle ghi \\) using only rigid motions.

Explanation:

Step1: Check side - length ratios

For \(\triangle DEF\) with side lengths \(9m\), \(13.5m\), \(15m\) and \(\triangle GHI\) with side lengths \(6m\), \(9m\), \(10m\).
Calculate the ratios: \(\frac{9}{6}=\frac{3}{2}\), \(\frac{13.5}{9}=\frac{3}{2}\), \(\frac{15}{10}=\frac{3}{2}\).

Step2: Analyze rigid - motion implications

Rigid motions (translation, rotation, reflection) preserve side lengths. Since the side lengths of \(\triangle DEF\) and \(\triangle GHI\) are not equal (they are in a ratio of \(\frac{3}{2}\), which means they are similar but not congruent). Rigid motions cannot change the size of a figure.

Answer:

No. It is not possible to map \(\triangle DEF\) onto \(\triangle GHI\) using only rigid motions.