Question

Georgia is justifying the SSS triangle congruence criteria using △ABC and △LMN (image 1). She transforms the first triangle with a translation to create △A′B′C′ (image 2), followed by a rotation to create △A"B"C" (image 3). She is surprised to see that △A"B"C" overlaps △LMN without having done a reflection. What happened? (1 point) Responses Georgia made a mistake. She should have done a reflection before the rotation so that her sequence of transformations included all three types of transformations. Georgia made a mistake. She should have done a reflection before the rotation so that her sequence of transformations included all three types of transformations. Georgia has shown that the two triangles are not congruent. If the sequence of transformations does not include a translation, rotation, and reflection, then the triangles cannot be congruent. Georgia has shown that the two triangles are not congruent. If the sequence of transformations does not include a translation, rotation, and reflection, then the triangles cannot be congruent. Georgia has shown that the two triangles are congruent. Sometimes three transformations are not necessary when mapping one triangle onto another triangle. Georgia has shown that the two triangles are congruent. Sometimes three transformations are not necessary when mapping one triangle onto another triangle. Georgia made a mistake. She should have translated △ABC along the vector that takes point B to point N, then done a reflection, then done a rotation.

Answer 1

Georgia has shown that the two triangles are congruent. Sometimes three transformations are not necessary when mapping one triangle onto another triangle.

In the context of triangle congruence, it is not necessary for all three types of transformations (translation, rotation, and reflection) to be used sequentially to demonstrate congruence. If the triangles overlap after her sequence of transformations (translation followed by rotation), that is sufficient evidence to show that they are congruent, since congruence means one triangle can be mapped onto another through a series of transformations.