Question

is (\triangle abc) congruent to (\triangle uvw)? justify your answer using rigid motions.
yes. a translation and a rotation can map (\triangle abc) onto (\triangle uvw).
yes. a translation and a reflection can map (\triangle abc) onto (\triangle uvw).
no. it is not possible to map (\triangle abc) onto (\triangle uvw) using only rigid motions.

Explanation:

Step1: Analyze the side - angle - side (SAS) correspondence

In \(\triangle ABC\), we have \(AB = 5\) in, \(AC=10\) in, \(BC = 12\) in. In \(\triangle UVW\), we have \(UV = 5\) in, \(UW = 10\) in, \(VW=12\) in.
For \(\triangle ABC\) and \(\triangle UVW\), if we consider the SAS (side - angle - side) congruence criterion. The angle in \(\triangle ABC\) is at \(A\) and the angle in \(\triangle UVW\) is at \(U\).

Step2: Determine the rigid motion

A translation can be used to move \(\triangle ABC\) so that one of its vertices coincides with a corresponding vertex of \(\triangle UVW\). But to make the angles match (since the angle in \(\triangle ABC\) and \(\triangle UVW\) are not in the same orientation initially), a reflection is needed.
A translation moves the triangle without rotation or reflection (preserves shape and size). A reflection flips the triangle over a line (also preserves shape and size).

Answer:

Yes. A translation and a reflection can map \(\triangle ABC\) onto \(\triangle UVW\).