Question

the triangles are congruent by the sss congruence theorem. which rigid transformation(s) can map △mnp onto △tsr? reflection only rotation only translation, then reflection translation, then rotation

Explanation:

Step1: Analyze reflection

Reflection alone cannot map \(\triangle MNP\) onto \(\triangle TSR\) as their orientations are not just mirror - images without considering position.

Step2: Analyze rotation

Rotation alone cannot map \(\triangle MNP\) onto \(\triangle TSR\) as their positions are not in a rotational - only relationship.

Step3: Analyze translation then reflection

First, translate \(\triangle MNP\) so that one of its vertices (say \(P\)) aligns with a corresponding vertex (say \(R\) or \(T\) after considering side lengths). Then, reflect the translated triangle. Since \(MP = RT=7\) and the other sides are equal (by SSS congruence), a translation (to align the base - like sides) followed by a reflection (to align the other sides and vertices) can map \(\triangle MNP\) onto \(\triangle TSR\).

Step4: Analyze translation then rotation

Translation then rotation will not map the sides and vertices correctly as the side - to - side and vertex - to - vertex correspondences (based on SSS congruence) are better achieved with translation then reflection.

Answer:

translation, then reflection