Question

georgia is justifying the sss triangle congruence criteria using \\( \triangle abc \\) and \\( \triangle lmn \\) (image 1). she transforms the first triangle with a translation to create \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) (image 2), followed by a rotation to create \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\) (image 3). she is surprised to see that \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\) overlaps \\( \triangle lmn \\) without having done a reflection. what happened? (1 point) 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 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 made a mistake. she should have translated \\( \triangle abc \\) along the vector that takes point \\( b \\) to point \\( n \\), then done a reflection, then done a rotation.

Explanation:

Georgia is using the SSS (Side - Side - Side) congruence criteria. Transformations like translation, rotation, and reflection are isometric transformations (they preserve the shape and size of the figure). If two triangles are congruent (by SSS in this case), there exists a sequence of isometric transformations (translation, rotation, and/or reflection) that can map one triangle onto the other. Just because she didn't use a reflection in her sequence of transformations (translation followed by rotation) does not mean the triangles are not congruent. In fact, if the two triangles are congruent (as per SSS), sometimes a reflection is not necessary. For example, if the orientation of the two triangles (after translation) is such that a rotation can map one onto the other without the need for a reflection.

Answer:

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