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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?

(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 \( EGH \) and show that the triangles are congruent, let's first analyze the coordinates for both triangles.

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

When we reflect triangle \( ABC \) over the x-axis, the y-coordinates of each vertex will change their signs.

  • Reflecting \( A(-6, -2) \):
    • New coordinates: \( A'(-6, 2) \)
  • Reflecting \( B(-4, -5) \):
    • New coordinates: \( B'(-4, 5) \)
  • Reflecting \( C(-1, -5) \):
    • New coordinates: \( C'(-1, 5) \)

Thus, after reflection over the x-axis, the new coordinates are:

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

Step 2: Translation

Next, we need to translate the reflected triangle \( A'B'C' \) to align it with triangle \( EGH \).

To find the translation, we examine the coordinates:

  • We can look at the vertex points where \( A'(-6, 2) \) should go to one of the vertices of triangle \( EGH \). We can use point \( H(2, 3) \) as a reference for the translation.

Translation vector

We need to move from \( A'(-6, 2) \) to \( H(2, 3) \):

  • The translation vector can be calculated by taking the coordinates of \( H \) and subtracting the coordinates of \( A' \):
    • \( \Delta x = 2 - (-6) = 2 + 6 = 8 \)
    • \( \Delta y = 3 - 2 = 1 \)

Thus, the translation vector is \( (8, 1) \).

Summary

The sequence of transformations needed to move triangle \( ABC \) to triangle \( EGH \) is:

  1. Reflection over the x-axis
  2. Translation by \( (8, 1) \)

So, the final answer for the transformations is: reflection over the x-axis, then a translation of (8, 1).