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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 negative 6 comma negative 2 right parenthesis, upper B left parenthesis negative 4 comma negative 5 right parenthesis, and upper C left parenthesis negative 1 comma negative 5 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis 4 comma 6 right parenthesis, upper G left parenthesis 7 comma 6 right parenthesis, and upper H left parenthesis 2 comma 3 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.

What is the sequence of transformations needed to move triangle ABC
to triangle EGH
to show that the triangles are congruent? reflection over the x
-axis, then a translation of

1 answer

To find the sequence of transformations needed to move triangle \( ABC \) to triangle \( EGH \) and to show that they are congruent, we can follow these steps:

  1. Reflection over the x-axis:

    • Reflect triangle \( ABC \) across the x-axis. This transformation changes the y-coordinates of each vertex of triangle \( ABC \) to their opposite signs, while the x-coordinates remain the same.

    • The new coordinates after this reflection will be:

      • A' (–6, 2)
      • B' (–4, 5)
      • C' (–1, 5)

  2. Translation:

    • Now we need to translate the reflected triangle \( A'B'C' \) so that it aligns with triangle \( EGH \).

    • To find the translation vector, we can look at where point A' (–6, 2) needs to go to match point E (4, 6).

    • The translation can be calculated as follows:

      • For the x-coordinates: \( 4 - (–6) = 4 + 6 = 10 \)
      • For the y-coordinates: \( 6 - 2 = 4 \)

    • Therefore, we translate triangle \( A'B'C' \) by \( (10, 4) \).

After performing these transformations, triangle \( ABC \) (after reflection and translation) would correspond to triangle \( EGH \).

Thus, the sequence of transformations is:

  1. Reflect triangle \( ABC \) over the x-axis.
  2. Translate the reflected triangle by \( (10, 4) \).

This proves that triangles \( ABC \) and \( EGH \) are congruent under these transformations.