Question

the triangles are congruent by the sss congruence theorem. which rigid transformation(s) can map △abc onto △fed? reflection, then dilation rotation, then reflection reflection, then translation rotation, then translation

Explanation:

Step1: Understand rigid transformations

Rigid transformations (rotation, reflection, translation) preserve the shape and size of the figure. Dilation is not a rigid transformation as it changes the size. So, the option with dilation (reflection, then dilation) is incorrect.

Step2: Analyze rotation - then - reflection

If we rotate \(\triangle ABC\) and then reflect it, we can map the vertices. Let's consider the orientation. Rotation can adjust the angle - orientation and reflection can adjust the mirror - like orientation.
For example, if we rotate \(\triangle ABC\) around a point (say the mid - point of \(AC\) and \(FD\) since \(AC = FD=20\)) to get the right - angle in a similar angular position as in \(\triangle FED\) and then reflect it (over a line perpendicular to the line joining the corresponding vertices), we can map \(A\) to \(F\), \(B\) to \(E\), and \(C\) to \(D\).

Step3: Analyze reflection - then - translation

If we just reflect \(\triangle ABC\), the orientation of the right - angle (in terms of the side - lengths around it) may not match. Translation only moves the figure without rotating or reflecting in a way that can correct the angular orientation of the right - angle with respect to the side - lengths.

Step4: Analyze rotation - then - translation

Translation alone (after rotation) cannot correct the mirror - like orientation (if there is one) between \(\triangle ABC\) and \(\triangle FED\). Rotation can handle the angular orientation but not the mirror - image aspect (if present) as effectively as a combination of rotation and reflection.

Answer:

rotation, then reflection