Question

1. 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 ghj and stu.
b translate triangle ghj by the directed line segment \\( \overline { g s } \\). rotate triangle ghj using s as the center by angle hst. reflect ghj over \\( \overline { s t } \\).
c translate triangle ghj by the directed line segment \\( \overline { g t } \\). rotate triangle ghj using t as the center by angle hts. reflect triangle ghj over \\( \overline { s t } \\).
d translate triangle ghj by the directed line segment \\( \overline { g s } \\). translate triangle ghj by the directed line segment ht. translate triangle ghj by the directed line segment jt.

Explanation:

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

• Option A: Rotating around \( G \) and then reflecting may not align the triangles properly as the initial rotation and reflection steps don't account for the side - side - side (SSS) congruence (given \( \overline{SU}\cong\overline{GJ},\overline{UT}\cong\overline{JH},\overline{TS}\cong\overline{HG} \)) in a consistent way.
• Option B:
• First, translating \( \triangle GHJ \) by \( \overrightarrow{GS} \) moves \( G \) to \( S \).
• Then, rotating the translated triangle \( G'H'J' \) about \( S \) by \( \angle H'ST \) aligns \( H' \) - related sides.
• Finally, reflecting over \( \overline{ST} \) will map the triangle to \( \triangle STU \) because the SSS congruence (from the given side - congruences) is maintained through these rigid motions (translation, rotation, reflection are all rigid motions that preserve shape and size).
• Option C: Translating by \( \overrightarrow{GT} \) does not start with aligning a vertex that is consistent with the SSS congruence setup, and the subsequent rotation and reflection steps will not correctly map \( \triangle GHJ \) to \( \triangle STU \).
• Option D: Multiple translations may not align the triangles correctly as translations alone can't account for the rotational or reflective aspects needed to match the SSS - congruent triangles.

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 \( \overline{ST} \).