Question

here are triangles ghj and stu.
\\( \overline { s u } \cong \overline { g j } \\)
\\( \overline { u t } \cong \overline { j h } \\)
\\( \overline { t s } \cong \overline { h g } \\)
which sequence of rigid motions will definitely work to take triangle ghj onto triangle stu?
a rotate triangle ghj using center g until \\( \overline { g h } \\) is lined up with \\( \overline { s t } \\). then reflect over a line halfway between \\( g ^ { prime } h ^ { prime } j ^ { prime } \\) and stu.
b translate triangle ghj by the directed line segment \\( \overline { g s } \\). rotate triangle \\( g ^ { prime } h ^ { prime } j ^ { prime } \\) using s as the center by angle \\( h ^ { prime } s t \\). reflect \\( g ^ { prime prime } h ^ { prime prime } j ^ { prime prime } \\) over \\( \overline { s t } \\).
c translate triangle ghj by the directed line segment \\( \overline { g t } \\). rotate triangle \\( g ^ { prime } h ^ { prime } j ^ { prime } \\) using t as the center by angle \\( h ^ { prime } t s \\). reflect triangle \\( g ^ { prime prime } h ^ { prime prime } j ^ { prime prime } \\) over \\( \overline { s t } \\).
d translate triangle ghj by the directed line segment \\( \overline { g s } \\). translate triangle \\( g ^ { prime } h ^ { prime } j ^ { prime } \\) by the directed line segment \\( h ^ { prime } t \\). translate triangle \\( g ^ { prime prime } h ^ { prime prime } j ^ { prime prime } \\) by the directed line segment \\( j ^ { prime prime } t \\).

Explanation:

To determine the correct sequence of rigid motions, we analyze each option:

• Option A: Rotating about \( G \) to line up \( \overline{GH} \) with \( \overline{ST} \) and then reflecting may not align all corresponding parts correctly, as the initial rotation center and subsequent reflection might not ensure congruence.
• Option B: Translating \( \triangle GHJ \) by \( \overrightarrow{GS} \) moves \( G \) to \( S \). Rotating about \( S \) by \( \angle H'ST \) aligns \( \overline{H'G'} \) (now at \( S \)) with \( \overline{ST} \), and reflecting over \( \overline{ST} \) ensures the triangles match, as the side - side - side (SSS) congruence (given \( \overline{SU}\cong\overline{GJ},\overline{UT}\cong\overline{JH},\overline{TS}\cong\overline{HG} \)) is maintained through these rigid motions.
• Option C: Translating by \( \overrightarrow{GT} \) does not target aligning the correct vertices first, so the subsequent rotation and reflection will not work as intended.
• Option D: Multiple translations do not account for the rotational or reflective alignment needed to match the triangles, as translations alone can't adjust the orientation to match \( \triangle STU \) from \( \triangle GHJ \).

Answer:

B. Translate triangle \( GHJ \) by the directed line segment \( \overrightarrow{GS} \). Rotate triangle \( G'H'J' \) using \( S \) as the center by angle \( H'ST \). Reflect \( G''H''J'' \) over \( \overline{ST} \).