Use the image to answer the question.

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Two triangles, upper A upper B upper C and upper E upper G upper H, are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis 1 comma negative 3 right parenthesis, upper B left parenthesis 1 comma negative 1 right parenthesis, and upper C left parenthesis 4 comma negative 3 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis negative 2 comma 5 right parenthesis, upper G left parenthesis negative 2 comma 2 right parenthesis, and upper H left parenthesis negative 4 comma 2 right parenthesis. In triangle upper E upper G upper H, side upper E upper G is labeled as h, side upper G upper H is labeled as e, and side upper H upper E is labeled as g.

△ABC is congruent to △GHE . Describe a sequence of transformations from △ABC to △GHE that proves their congruence.

1 answer

To prove that triangle ABC is congruent to triangle GHE, we can use the sequence of transformations:

1. Translation: Move triangle ABC two units to the right and three units up. This will result in a new triangle with vertices A(3, 0), B(3, 2), and C(6, 0).

2. Reflection: Reflect the translated triangle over the x-axis. This will result in a new triangle with vertices A(3, 0), B(3, -2), and C(6, 0).

3. Rotation: Rotate the reflected triangle 180 degrees counterclockwise about the origin. This will result in a new triangle with vertices E(-3, 0), G(-3, 2), and H(-6, 0).

The transformed triangle EGH, obtained after the translation, reflection, and rotation, is congruent to triangle ABC. Therefore, triangles ABC and GHE are congruent.