Question

which rigid transformation(s) can map (\triangle mnp) onto (\triangle tsr)?
rotation only
translation, then reflection
reflection only
translation, then rotation

Explanation:

Step1: Analyze the orientation of the triangles

$\triangle MNP$ and $\triangle TSR$ have the same side - length (by SSS congruence, $MP = RT=7$, and other corresponding sides are equal as marked). But their orientations are different. A rotation alone cannot map $\triangle MNP$ onto $\triangle TSR$ because of the non - matching vertex order in terms of orientation. A reflection alone also won't work as the position (in terms of translation - like offset) is not accounted for.

Step2: Consider translation and rotation

First, translate $\triangle MNP$ so that one of its vertices (say, the vertex corresponding to the base of length 7) is in the same position as the corresponding vertex of $\triangle TSR$. Then, rotate the translated $\triangle MNP$ around the appropriate point (the common vertex after translation) to align all the vertices. A translation moves the triangle without rotation or reflection, and then a rotation can adjust the orientation.

Answer:

translation, then rotation