Question

tammy was curious if quadrilaterals abcd and fghe were congruent. she was able to map one figure onto the other using a reflection and a translation. tammy concluded: \i was able to map quadrilateral abcd onto fghe using a sequence of rigid transformations, so the figures are congruent.\ what error did tammy make in her conclusion?

Explanation:

Rigid transformations (reflections, translations, rotations) preserve the shape and size of a figure, meaning the pre - image and image are congruent. However, Tammy's error is in the mapping direction or the identification of the figures. The rigid transformations she used (reflection and translation) were likely to map one figure to the other, but the key is that congruence via rigid transformations requires that the transformations map the vertices correctly. In this case, the error is that the sequence of transformations (reflection and translation) does not map ABCD onto FGHE correctly. The labels of the vertices should correspond after the transformation. When using rigid transformations, the order of the vertices (the correspondence) must be maintained. Tammy likely misidentified the correspondence of the vertices. For two quadrilaterals to be congruent via rigid transformations, the transformation should map \(A\) to \(F\), \(B\) to \(G\), \(C\) to \(H\), \(D\) to \(E\) (or the correct vertex - to - vertex correspondence) after the reflection and translation. But from the diagram, the reflection and translation as described do not result in the correct vertex correspondence. So the error is that the sequence of rigid transformations (reflection and translation) does not map quadrilateral \(ABCD\) onto \(FGHE\) (the vertex correspondence is incorrect), so her conclusion that she mapped \(ABCD\) onto \(FGHE\) is wrong.

Answer:

Tammy's error is that the sequence of rigid transformations (reflection and translation) does not actually map quadrilateral \(ABCD\) onto \(FGHE\) (the vertex correspondence is incorrect), yet she claims she did so to conclude congruence.