The coordinate plane shows the locations of triangle 1 and triangle 2.

Question

The triangle 1 has vertices at (-4, 2), (-5, 5), and (-1, 4). The tringle 2 has vertices at (2, -4), (5, -5), and (4, -1).

Jessica made the following statement.

Triangle 2 is the result of a specific rigid motion: a rotation of triangle 1 about the origin. The result is two triangles that are similar to one another but not congruent.
Which of these best describes Jessica's statement?

A.
She made an incorrect statement. The two triangles are not similar because the angles for triangle 2 are not the same as for triangle 1.

B.
She made two errors. The rigid motion is a reflection not a rotation and the two triangles are both similar and congruent.

C.
She made a correct statement.

D.
She called the transformation a rigid motion, when actually it is not. In addition, she describes triangle 2 as being a rotation, when it is the result of a reflection. Also, the two triangles are both similar and congruent.

Answer 1

To evaluate Jessica's statement, we need to consider two main points: the transformation from triangle 1 to triangle 2, and whether the triangles are similar or congruent.

1. Transformation: If triangle 2 is indeed the result of a rigid motion (which includes rotations, reflections, and translations), the angles and side lengths must remain unchanged. To confirm if it is a rotation or reflection, we can analyze the coordinates of the triangles.

• Triangle 1 vertices: (-4, 2), (-5, 5), (-1, 4)
• Triangle 2 vertices: (2, -4), (5, -5), (4, -1)

To determine if triangle 2 is a rotation of triangle 1 about the origin, we would check if the coordinates of triangle 1 can be produced by rotating triangle 2. A rotation (counter-clockwise) around the origin (x, y) to (y, -x) or similar transformations typically involve checking for specific angle movements. Given that the coordinates change from negative to positive and vice versa, it suggests a possible reflection rather than a simple rotation.

2. Similar or Congruent: Triangles are similar if their corresponding angles are equal and corresponding sides are proportional. They are congruent if all corresponding sides and angles are equal.

From the coordinates provided, it is necessary to either calculate the side lengths and angles or observe the proportional relationships visually to determine similarity. However, given the statements, the suggested outcome implies mismatched angles and lengths.

Conclusion:

Given this analysis, Jessica seems to mischaracterize the transformation and the similarity of the triangles. Therefore, the best descriptor of Jessica's statement is:

D. She called the transformation a rigid motion, when actually it is not. In addition, she describes triangle 2 as being a rotation, when it is the result of a reflection. Also, the two triangles are both similar and congruent.

Though it is possible they might be congruent depending on the outcome of side or angle analysis, confirming the exact relationships generally requires further calculations or geometrical proof. The key point, however, is that she mischaracterizes the transformation and the relationship between the triangles.