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

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

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

rotation 90 degrees clockwise, then a translation of (1, 6)

Answer 1

To determine the correct sequence of steps to move triangle ABC in Quadrant IV to triangle EGH in Quadrant II, we need to analyze the positions and movements of the triangles.

1. Coordinates of Triangle ABC:

   • A (1, -1)
   • B (1, -3)
   • C (4, -3)

2. Coordinates of Triangle EGH:

   • E (-2, 5)
   • G (-2, 3)
   • H (-5, 3)

To move triangle ABC to match triangle EGH, we can follow these steps:

1. Reflection: Triangle ABC, located in Quadrant IV, can be reflected over the x-axis. This will move the points to Quadrant III:

   • A' (1, 1)
   • B' (1, 3)
   • C' (4, 3)

2. Translation: After reflection, we need to translate triangle A'B'C' to the coordinates of triangle EGH. By translating the reflected triangle up and left, we can align the triangles properly.

To evaluate the options:

• Reflection to Quadrant III, then a translation of (−1, 6) matches our steps as A' needs to translate to E by moving left and up after the reflection.

Thus, the correct response is reflection to Quadrant III, then a translation of (−1, 6).