Use the image to answer the question..

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)

Answer 1

To move triangle ABC in Quadrant IV to triangle EGH in Quadrant II and show that the triangles are congruent, we need to follow these steps:

1. Reflect triangle ABC across the y-axis to get triangle A'B'C' in Quadrant III. This turns the coordinates of triangle ABC from:

   • A(1, -1) to A'(-1, -1)
   • B(1, -3) to B'(-1, -3)
   • C(4, -3) to C'(4, -3) (the x-coordinate changes signs).

2. From Quadrant III, we can then translate triangle A'B'C' to Quadrant II. The translation can be defined as moving by (-1, 6) which would move points up and left as follows:

   • A'(-1, -1) to A''(-1 - 1, -1 + 6) = (-2, 5)
   • B'(-1, -3) to B''(-1 - 1, -3 + 6) = (-2, 3)
   • C'(4, -3) to C''(4 - 1, -3 + 6) = (-5, 3)

This matches the vertices of triangle EGH.

Thus, the correct response is:

reflection to Quadrant III, then a translation of (−1, 6).