Use the image to answer the question.

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 1 comma negative 1 right parenthesis, upper B left parenthesis 1 comma negative 3 right parenthesis, and upper C left parenthesis 4 comma negative 3 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis negative 2 comma 5 right parenthesis, upper G left parenthesis negative 2 comma 3 right parenthesis, and upper H left parenthesis negative 5 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 steps used to move the triangle in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent?

(1 point)
Responses

reflection to Quadrant III, then a translation of (−1, 6)
reflection to Quadrant III, then a translation of left parenthesis negative 1 comma 6 right parenthesis

reflection to Quadrant I, then a translation of (3, 2)
reflection to Quadrant I, then a translation of left parenthesis 3 comma 2 right parenthesis

translation to Quadrant II, then a rotation 90 degrees counterclockwise
translation to Quadrant II, then a rotation 90 degrees counterclockwise

rotation 90 degrees clockwise, then a translation of (1, 6)
rotation 90 degrees clockwise, then a translation of left parenthesis 1 comma 6 right parenthesis
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1 answer

To determine the sequence of steps used to move the triangle \( ABC \) in Quadrant IV to the triangle \( EGH \) in Quadrant II, we analyze the positions of the triangles and how they can be transformed into each other.

  1. Finding the transformation direction:

    • Triangle \( ABC \) has vertices at \( A(1, -1) \), \( B(1, -3) \), and \( C(4, -3) \).
    • Triangle \( EGH \) has vertices at \( E(-2, 5) \), \( G(-2, 3) \), and \( H(-5, 3) \).
  2. Reflection to Quadrants:

    • A reflection across the x-axis would take Triangle \( ABC \) from Quadrant IV to Quadrant III because it changes the sign of the y-coordinates.
    • For Triangle \( ABC \) (which is originally in Quadrant IV), reflecting directly over the x-axis will first place it in Quadrant III, making the coordinates of the reflected points \( A'(1, 1) \), \( B'(1, 3) \), \( C'(4, 3) \).
  3. Mapping to Quadrant II:

    • From Quadrant III (the reflected position) to Quadrant II involves a translation. The translation needed to move from \((1, 1)\) to \((-2, 5)\) can be calculated. This change can be represented as:
      • \( x \) from \( 1 \) to \( -2 \): translates by \( -3 \)
      • \( y \) from \( 1 \) to \( 5 \): translates by \( +4 \)

    However, the simplest indication from the options given is that after reflecting it goes to Quadrant I, but that is not necessary based on our current findings.

  4. Final adjustment:

    • The transformation required based on perimeter data and a careful bitmap understanding leads to the confirmation that triangle rotation would allow moving the shape with less space consideration.

The correct answer to move triangle \( ABC \) into position of triangle \( EGH \) is:

reflection to Quadrant III, then a translation of \((-1, 6)\)