Question

determining a combination of rigid transformations
how can (\triangle abc) be mapped to (\triangle xyz)?
first, translate vertex a to vertex.
next, (\triangle abc) to align the sides and angles.

Explanation:

Step1: Analyze the correspondence of vertices

In rigid - transformations (translations, rotations, reflections), we want to map one triangle to another. For \(\triangle ABC\) and \(\triangle XYZ\), the side lengths \(AB = XY=28\mathrm{cm}\) and \(AC = XZ = 32\mathrm{cm}\). The angle between \(AB\) and \(AC\) in \(\triangle ABC\) corresponds to the angle between \(XY\) and \(XZ\) in \(\triangle XYZ\). So, vertex \(A\) should be translated to vertex \(X\).

Step2: Determine the second transformation

After translating vertex \(A\) to vertex \(X\), we need to rotate \(\triangle ABC\). Rotation is a transformation that turns a figure around a fixed point. Since the orientation of the triangles (the way the sides and angles are arranged) is different (if we assume the initial translation of \(A\) to \(X\) only moves the position but not the orientation), a rotation will align the sides and angles of \(\triangle ABC\) with \(\triangle XYZ\).

Answer:

First, translate vertex \(A\) to vertex \(X\). Next, rotate \(\triangle ABC\) to align the sides and angles.