Question

think about the process how can you decide if \\( \triangle def \cong \triangle def \\)? is \\( \triangle def \cong \triangle def \\)? explain.
is \\( \triangle def \cong \triangle def \\)? explain. choose the correct answer below.
\\( \circ \\) a. yes, because a rotation of \\( 180^{\circ} \\) about the origin, followed by translations of 3 units down and 4 units left map \\( \triangle def \\) to \\( \triangle def \\).
\\( \circ \\) b. yes, because a rotation of \\( 180^{\circ} \\) about the origin, followed by translations of 4 units up and 3 units right map \\( \triangle def \\) to \\( \triangle def \\).
\\( \circ \\) c. yes, because translations of 4 units down and 3 units left, followed by a rotation of \\( 90^{\circ} \\) about the origin and a reflection across the x - axis map \\( \triangle def \\) to \\( \triangle def \\).
\\( \circ \\) d. no, because a sequence of rigid motions does not map \\( \triangle def \\) to \\( \triangle def \\).

Explanation:

Step1: Recall Rigid Motion Properties

Rigid motions (rotations, translations, reflections) preserve congruence. We analyze each option's transformation sequence.

Step2: Analyze Option A

A 180° rotation about the origin changes coordinates \((x,y)\) to \((-x,-y)\). Then translating 3 units down (subtract 3 from \(y\)) and 4 units left (subtract 4 from \(x\)) would map \(\triangle DEF\) to \(\triangle D'E'F'\) if their coordinates match after transformation. Checking the graph, this sequence aligns with the positions of the triangles, preserving congruence.

Step3: Eliminate Other Options

• Option B: Rotation 180° + translation 4 up/3 right doesn't match the triangle positions.
• Option C: Translations 4 down/3 left + 90° rotation + reflection doesn't align with the congruence - preserving path.
• Option D: Incorrect, as rigid motions (rotation + translation) do map them, so congruence holds.

Answer:

A. Yes, because a rotation of \(180^\circ\) about the origin, followed by translations of 3 units down and 4 units left map \(\triangle DEF\) to \(\triangle D'E'F'\).