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?

Answer 1

In this scenario, Georgia is using a combination of a translation and a rotation to create a new triangle (△A"B"C") that overlaps with triangle △LMN. The fact that the two triangles overlap without any reflection is an important observation that can be understood in terms of triangle properties and congruence.

Here’s what happened:

1. Translation: When Georgia translated △ABC to create △A′B′C′, she moved the entire triangle to a new position in the plane. This action preserved the lengths of the sides and the measures of the angles, meaning △A′B′C′ is congruent to △ABC.

2. Rotation: After translation, Georgia then rotated the triangle (A′B′C′) to form triangle A"B"C". A rotation also preserves side lengths and angle measures. Therefore, △A"B"C" is still congruent to △ABC and thus to △A′B′C′.

Since both translation and rotation are rigid transformations, the resulting triangle, △A"B"C", is congruent to the original triangle, △ABC, without any change to its size or shape. When △A"B"C" overlaps with △LMN, it suggests that:

• Triangle △LMN is also congruent to triangle △ABC (which means they have the same side lengths and angles).

In this case, there was no need for a reflection because congruence can be established purely through translations and rotations. This points to the fact that triangle △LMN may have been located at the position of triangle △A"B"C" after the transformations, but the shapes remain congruent.

To sum up, Georgia's conclusion about the congruence between triangles is valid because congruence can arise from any combination of translations, rotations, and reflections. The overlap is simply a result of the specific transformations applied to triangle △ABC.