Question

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

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

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

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

Explanation:

Step1: Recall the properties of rigid motions

Rigid motions (translations, rotations, reflections) preserve side - lengths and angles.

Step2: Compare the side - lengths of the two triangles

In \(\triangle DEF\), the side - lengths are \(DE = 5\mathrm{cm}\), \(DF=12\mathrm{cm}\), \(EF = 13\mathrm{cm}\). In \(\triangle XYZ\), the side - lengths are \(XY = 9\mathrm{cm}\), \(XZ = 12\mathrm{cm}\), \(YZ=15\mathrm{cm}\). Since \(DE
eq XY\) and \(EF
eq YZ\), the two triangles do not have the same set of side - lengths.

Answer:

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