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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)

1 answer

To demonstrate the congruence of triangles \( ABC \) and \( GHE \), we can describe a sequence of transformations. Here are the steps:

  1. Translation: First, translate triangle \( ABC \) so that point \( A(1, -3) \) moves to point \( E(-2, 5) \). This requires moving \( A \) left by 3 units and up by 8 units. Therefore, triangle \( ABC \) is translated by the vector \( (-3, 8) \).

  2. Reflection: After the translation, reflect the translated triangle across the line \( y = 2 \). This line is horizontal, and reflecting across it will flip the positions of points vertically.

  3. Rotation (if needed): If the position of triangle \( ABC \) after the previous transformations does not match triangle \( GHE \) in terms of orientation, perform a rotation around the centroid of the triangles or around one of the points.

These transformations—translation, reflection (and possibly rotation)—will ensure that triangle \( ABC \) overlaps exactly with triangle \( GHE \), proving their congruence through the sequence of transformations.