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.

Question

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.

Answer 1

To find the sequence of transformations needed to move triangle ABC to triangle EGH and show that the triangles are congruent, let's analyze the coordinates and the necessary transformations step-by-step.

1. Reflection: We need to reflect triangle ABC over the x-axis. This will change the y-coordinates of the vertices of triangle ABC from negative values to positive values:

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

   After reflecting, the coordinates of triangle A'B'C' are:

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

2. Translation: Next, we need to translate triangle A'B'C' to match triangle EGH. We will find the translation vector needed to move from A' to E, since E is one of the vertices of triangle EGH:

   • The vertex E is located at (4, 6) and A' is at (-6, 2).
   • To find the translation vector, we calculate:
     • Change in x = 4 - (-6) = 4 + 6 = 10
     • Change in y = 6 - 2 = 4
   • Therefore, the translation vector is (10, 4).

The complete sequence of transformations is:

1. Reflect triangle ABC over the x-axis.
2. Translate the reflected triangle A'B'C' by the vector (10, 4) to obtain triangle EGH.

Thus, the transformations needed to move triangle ABC to triangle EGH are: reflection over the x-axis, then a translation of (10, 4).