Use the image to answer the question.

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.

(2 points)

Math Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is correct. The student made no errors while solving.

The answer shows that the student used problem-solving skills and reasoning.

All work is shown or explained.

1
The answer is partially correct. The student made errors while solving.

The answer shows little problem-solving skills or reasoning.

Some work is shown or explained.

0 The question is not answered.

Answer 1

To demonstrate the congruence of triangles \( \triangle ABC \) and \( \triangle GHE \), we can describe the following sequence of transformations:

1. Translation: Move triangle \( \triangle ABC \) so that point \( A(1, -3) \) coincides with point \( E(-2, 5) \). This requires translating triangle \( ABC \) left by 3 units (from \( x = 1 \) to \( x = -2 \)) and up by 8 units (from \( y = -3 \) to \( y = 5 \)).

2. Reflection: Reflect the translated triangle across the line \( x = -2 \) to account for the location of points \( E \), \( G \), and \( H \). This will map the new positions of points to the correct alignment with triangle \( GHE \).

3. Rotation (if necessary): If after reflection the orientation of the triangle is not aligned with \( \triangle GHE \), a rotation can be used about the appropriate point (this may not be necessary depending on the reflection).

These transformations preserve the side lengths and angles, ensuring that \( \triangle ABC \) and \( \triangle GHE \) are congruent.

In summary, the sequence of transformations to show congruence is:

1. Translate \( \triangle ABC \) left 3 units and up 8 units.
2. Reflect across the line \( x = -2 \).
3. (If necessary) Rotate to align orientation.

This shows that \( \triangle ABC \) can be transformed into \( \triangle GHE \), proving their congruence.