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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 ABCto triangle EGHto show that the triangles are congruent?(1 point)reflection over the x-axis, then a translation of ()

1 answer

To determine the sequence of transformations needed to move triangle ABC to triangle EGH, we will analyze their positions on the coordinate plane.

Step 1: Identifying Vertices

  • Triangle ABC vertices:

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

  • Triangle EGH vertices:

    • E (4, 6)
    • G (7, 6)
    • H (2, 3)

Step 2: Reflection over the x-axis

To reflect triangle ABC over the x-axis, we will change the signs of the y-coordinates of each vertex:

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

After reflecting triangle ABC across the x-axis, the new vertices will be:

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

Step 3: Translate the reflected triangle

Now we need to translate the new triangle A'B'C' so it aligns with triangle EGH.

We can observe the position of vertices A', B', and C' and find a translation vector:

  • Reach from A' (-6, 2) to E (4, 6):
    • x: 4 - (-6) = 10
    • y: 6 - 2 = 4
  • So the translation vector is (10, 4).

Conclusion

The sequence of transformations to move triangle ABC to triangle EGH is:

  1. Reflection over the x-axis
  2. Translation of (10, 4)

Thus, the answer is:

Reflection over the x-axis, then a translation of (10, 4).