Ask a New Question
Menu

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 (
)

2 answers

To determine the sequence of transformations needed to move triangle ABC to triangle EGH and show that the triangles are congruent, we first need to analyze their coordinates.

  • Triangle ABC:

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

  • Triangle EGH:

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

Step 1: Reflection over the x-axis

Reflecting triangle ABC over the x-axis changes the y-coordinates of each vertex to their opposites:

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

After this reflection, the new coordinates of triangle A'B'C' are:

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

Step 2: Translation to move A'B'C' to EGH

Now, we need to translate triangle A'B'C' so that vertices A'(-6, 2), B'(-4, 5), and C'(-1, 5) correspond with vertices E(4, 6), G(7, 6), and H(2, 3).

To find the translation vector, we can use the coordinates of one of the points. Let's translate A'(-6, 2) to E(4, 6).

The translation needed to move from A'(-6, 2) to E(4, 6) can be calculated as follows:

  • For the x-coordinate: 4 - (-6) = 4 + 6 = 10
  • For the y-coordinate: 6 - 2 = 4

Thus, we need to translate the triangle by the vector (10, 4).

Conclusion

Put together, the transformations to move triangle ABC to triangle EGH are:

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

So, the final answer in your requested format is: Reflection over the x-axis, then a translation of (10, 4).

no pick other one