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
△
A
B
C
is congruent to △GHE
△
G
H
E
. Describe a sequence of transformations from △ABC
△
A
B
C
to △GHE
△
G
H
E
that proves their congruence.

Answer 1

To show that triangle \( ABC \) is congruent to triangle \( GHE \) through a sequence of transformations, we can follow these steps:

1. Translation: First, we can translate triangle \( ABC \) so that it moves to the left along the x-axis and upwards along the y-axis. For example, translating triangle \( ABC \) left by 3 units (to the left) and up by 3 units (to the top) would mean adding \( (-3, 3) \) to each of the vertices of triangle \( ABC \):
   • Point \( A(1, -3) \) translates to \( A'(-2, 0) \)
   • Point \( B(1, -1) \) translates to \( B'(-2, 2) \)
   • Point \( C(4, -3) \) translates to \( C'(1, 0) \)

The resulting vertices \( A' \) and \( B' \) are positioned relative to points \( G \) and \( H \).

2. Reflection: Next, we can reflect the translated triangle \( A'B'C' \) across the vertical line \( x = -2 \) (or the y-axis), which will flip the triangle to match the orientation and position of triangle \( GHE \).
   • After the reflection, the points would match with those of triangle \( GHE \).

By applying these transformations—translation followed by reflection—triangle \( ABC \) can be transformed to triangle \( GHE \). This series of transformations demonstrates the congruence of the two triangles.

Therefore, the transformations that show \( \triangle ABC \cong \triangle GHE \) can be summarized as follows:

1. Translate triangle \( ABC \) left by 3 units and up by 3 units.
2. Reflect the resulting triangle across the line \( x = -2 \).

These transformations will align triangle \( ABC \) with triangle \( GHE \), proving their congruence.