Triangles A B C and D E F are shown. Triangle A B C is rotated to the left about point A and then is shifted up and to the right to form triangle D E F.

Question

What are the rigid transformations that will map
△ABC to △DEF?

Translate vertex A to vertex D, and then reflect
△ABC across the line containing AC.
Translate vertex B to vertex D, and then rotate
△ABC around point B to align the sides and angles.
Translate vertex B to vertex D, and then reflect
△ABC across the line containing AC.
Translate vertex A to vertex D, and then rotate
△ABC around point A to align the sides and angles.

Answer 1

To determine the rigid transformations that will map triangle $\mathrm{△}ABC$ to triangle $\mathrm{△}DEF$, we need to analyze the transformations described in the options provided.

1. Translate vertex A to vertex D, and then reflect $\mathrm{△}ABC$ across the line containing AC.

   • This transformation first translates $A$ to $D$ and then reflects, which might not align $\mathrm{△}ABC$ with $\mathrm{△}DEF$'s orientation.

2. Translate vertex B to vertex D, and then rotate $\mathrm{△}ABC$ around point B to align the sides and angles.

   • Translating $B$ to $D$ is incorrect since we need to move $A$ to $D$ to maintain the triangle structure.

3. Translate vertex B to vertex D, and then reflect $\mathrm{△}ABC$ across the line containing AC.

   • Similar to option 2, translating $B$ to $D$ isn't a fit for our required transformation.

4. Translate vertex A to vertex D, and then rotate $\mathrm{△}ABC$ around point A to align the sides and angles.

   • This transformation starts with translating $A$ to $D$, which is appropriate, and rotating around $A$ can help align triangle $ABC$ with triangle $DEF$ correctly.

Considering all options, the most appropriate transformation that correctly matches the description of moving triangle $\mathrm{△}ABC$ to $\mathrm{△}DEF$ is:

Translate vertex A to vertex D, and then rotate $\mathrm{△}ABC$ around point A to align the sides and angles.