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.

Explanation:

Step1: Analyze translation

Translate vertex \(A\) to vertex \(D\). This is the first - step in moving \(\triangle ABC\) towards the position of \(\triangle DEF\) as we want to match the starting point of the transformation.

Step2: Analyze rotation

After translating vertex \(A\) to vertex \(D\), rotate \(\triangle ABC\) around point \(A\) (now coinciding with \(D\)) to align the sides and angles. Since rigid transformations (translation and rotation) preserve the shape and size of the triangle, this will map \(\triangle ABC\) to \(\triangle DEF\).

Let's check other options:

• For the option of translating \(A\) to \(D\) and then reflecting across \(AC\): Reflection across \(AC\) will not align the triangles as needed.
• For the options of translating \(B\) to \(D\): The correspondence of vertices in \(\triangle ABC\) and \(\triangle DEF\) ( \(A\) corresponds to \(D\), \(B\) corresponds to \(E\), \(C\) corresponds to \(F\)) makes starting the translation from \(B\) incorrect.

Answer:

Translate vertex \(A\) to vertex \(D\), and then rotate \(\triangle ABC\) around point \(A\) to align the sides and angles.