Question

is \\( \triangle a b c \\) congruent to \\( \triangle d e f \\)? justify your answer using rigid motions.
yes. a translation and a reflection can map \\( \triangle a b c \\) onto \\( \triangle d e f \\).
yes. a translation and a rotation can map \\( \triangle a b c \\) onto \\( \triangle d e f \\).
no. it is not possible to map \\( \triangle a b c \\) onto \\( \triangle d e f \\) using only rigid motions.

Explanation:

Rigid motions (translation, rotation, reflection) preserve side - lengths. In \(\triangle ABC\), the side - lengths are \(AB = 12m\), \(BC=16m\), \(AC = 14m\). In \(\triangle DEF\), the side - lengths are \(DE = 12m\), \(EF = 20m\), \(FD=18m\). Since \(BC
eq EF\) (\(16
eq20\)) and \(AC
eq FD\) (\(14
eq18\)), the two triangles do not have the same side - lengths. Rigid motions cannot map a triangle with side - lengths \(12\), \(14\), \(16\) to a triangle with side - lengths \(12\), \(18\), \(20\) because rigid motions preserve the lengths of the sides of a figure.

Answer:

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