Question

triangles \\(\delta def\\) and \\(\delta abc\\) are congruent.

which of the following sequences of transformations maps \\(\delta def\\) onto \\(\delta abc\\)?

sequence a
a translation along the directed line segment \\(\overline{db}\\), then a \\(90^{\circ}\\) rotation about point \\(b\\).

sequence b
a \\(90^{\circ}\\) rotation about point \\(f\\), then a translation along the directed line segment \\(\overline{fc}\\).

Explanation:

Identify corresponding vertices of the congruent triangles

Using the Geometric Transformations knowledge point, we determine the mapping between the congruent triangles \(\Delta DEF\) and \(\Delta ABC\).
By comparing the orientation and side lengths:

• Vertex \(D\) corresponds to vertex \(B\).
• Vertex \(E\) corresponds to vertex \(C\).
• Vertex \(F\) corresponds to vertex \(A\).

Evaluate Sequence A

Using the Translations in Geometry and Rotations in Geometry knowledge points:

1. Translate \(\Delta DEF\) along the directed line segment \(\overline{DB}\). This maps vertex \(D\) directly onto vertex \(B\). Let the intermediate triangle be \(\Delta D'E'F'\), where \(D'\) is at \(B\).
2. Rotate \(\Delta D'E'F'\) by \(90^\circ\) about point \(B\) (which is \(D'\)).

• In \(\Delta DEF\), the segment \(DE\) is vertical (pointing downwards from \(D\)).
• In \(\Delta ABC\), the corresponding segment \(BC\) goes downwards and to the right.
• A counterclockwise rotation of \(90^\circ\) about \(B\) of a vertical downward segment \(D'E'\) would point to the right, which does not align with \(BC\).
• A clockwise rotation of \(90^\circ\) about \(B\) would point to the left.
• Thus, Sequence A does not map \(\Delta DEF\) onto \(\Delta ABC\).

Evaluate Sequence B

Using the Rotations in Geometry and Translations in Geometry knowledge points:

1. Rotate \(\Delta DEF\) by \(90^\circ\) counterclockwise about point \(F\).

• The vertical segment \(DE\) (length 4 units) is rotated.
• The horizontal distance from \(E\) to \(F\) is 2 units, and vertical is 4 units.
• Rotating \(\Delta DEF\) by \(90^\circ\) counterclockwise about \(F\) reorients the triangle so that the segment corresponding to \(DE\) becomes horizontal, matching the orientation of \(AB\) in \(\Delta ABC\).

1. Translate the rotated triangle along the directed line segment \(\overline{FC}\).

• Since \(F\) corresponds to \(A\), rotating about \(F\) keeps \(F\) in place.
• Translating along \(\overline{FC}\) would map the image of \(F\) (which is at \(F\)) to \(C\).
• However, \(F\) must map to \(A\), not \(C\).
• Thus, Sequence B does not map \(\Delta DEF\) onto \(\Delta ABC\).

Conclusion

Neither sequence correctly maps \(\Delta DEF\) onto \(\Delta ABC\).

Answer:

• (A) Sequence A only
• (B) Sequence B only
• (C) Both Sequence A and Sequence B
• (D) Neither sequence (Correct answer)