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 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. Georgia made a mistake. She should have translated triangle upper A upper B upper C along the vector that takes point upper B to point upper N , then done a reflection, then done a rotation. 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.

Question

the photos:
Diagram (1)
Triangle ABC: This is the original triangle with vertices labeled A, B, and C.
Segments BM and AL: These segments are drawn inside the triangle, intersecting at point N.
Shaded Area: The region where segments BM and AL intersect is shaded.
Diagram (2)
Triangle ABC: The same triangle as in Diagram (1) with the same internal segments BM and AL intersecting at N.
Extended Triangle A’B’C’: A new point C’ is added on the extension of side AB, forming a larger triangle.
Extended Segments: Segment BM is extended through M to meet side AC’ at M’.
Shaded Area: Now includes triangles ABN and ANM’.
Diagram (3)
Duplicated Triangle A"B"C": Triangle ABC is duplicated next to the original, forming triangle A"B"C".
Extended Segments: Segments BL, BM, and AL are extended to meet sides of the new triangle A"B"C" at points L’, M’, and L" respectively.
Shaded Area: Includes triangles ABN, ANM’, AM’L’, and AL’B.
The diagrams illustrate a sequence of transformations and constructions within and around triangle ABC, showing how segments and points are extended and intersected to form new geometric figures and shaded regions.

Answer 1

Georgia's conclusion that the triangles are congruent is valid, and it is important to note that congruence can be established through various sequences of transformations. Specifically, the correct response is:

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

The key idea is that while translation, rotation, and reflection are indeed the three rigid transformations used to prove congruence, it is not always necessary to utilize all three to establish that two triangles are congruent. Transformation sequences may vary, and it's possible for triangles to overlap congruently through less than the full set of transformations. Thus, even without explicitly performing a reflection, if transformations through translation and rotation result in the two triangles overlapping perfectly, then they are congruent.