Question

is there a series of rigid transformations that could map δqrs to δabc? if so, which transformations could be used? ○ no, δqrs and δabc are congruent but δqrs cannot be mapped to δabc using a series rigid transformations. ○ no, δqrs and δabc are not congruent. ○ yes, δqrs can be translated so that r is mapped to b and then rotated so that s is mapped to c. ○ yes, δqrs can be translated so that q is mapped to a and then reflected across the line containing qs.

Explanation:

Step1: Check congruence

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle QRS\) and \(\triangle ABC\) are congruent. \(QR = AB=16\mathrm{cm}\), \(RS = BC = 24\mathrm{cm}\), \(\angle QRS=\angle ABC = 90^{\circ}\)

Step2: Analyze rigid transformations

A rigid transformation is a transformation that preserves the shape and size of a figure. A translation moves a figure without rotating or resizing it, and a reflection flips a figure over a line.
If we translate \(\triangle QRS\) so that \(Q\) is mapped to \(A\) (translation moves the triangle to the correct position horizontally/vertically) and then reflect \(\triangle QRS\) across the line containing \(\overline{QS}\) (reflection changes the orientation to match \(\triangle ABC\))

Answer:

Yes, \(\triangle QRS\) can be translated so that \(Q\) is mapped to \(A\) and then reflected across the line containing \(\overline{QS}\).