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

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

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

Answer 1

To move triangle ABC in Quadrant IV to triangle EGH in Quadrant II and to show that these triangles are congruent, we need to consider the positions of the triangles and the geometric transformations that will align them.

1. Triangle ABC has vertices at A(1, -1), B(1, -3), and C(4, -3) in Quadrant IV.
2. Triangle EGH has vertices at E(-2, 5), G(-2, 3), and H(-5, 3) in Quadrant II.

To transform triangle ABC so that it matches triangle EGH, we can first reflect it over the y-axis (which moves it to Quadrant III) and then further translate it to position in Quadrant II.

Reflecting Triangle ABC:

• The reflection of A(1, -1) over the y-axis is A'(-1, -1).
• The reflection of B(1, -3) over the y-axis is B'(-1, -3).
• The reflection of C(4, -3) over the y-axis is C'(-4, -3).

Now, triangle A'B'C' in Quadrant III can be translated to match the positions of EGH in Quadrant II:

Translating the reflected triangle A'B'C':

• Translation of A'(-1, -1) by (-1, 6) will give us E(-2, 5).
• Translation of B'(-1, -3) by (-1, 6) will give us G(-2, 3).
• Translation of C'(-4, -3) by (-1, 6) will give us H(-5, 3).

Based on this analysis, the correct sequence of transformations to take triangle ABC from Quadrant IV to triangle EGH in Quadrant II is:

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

So the answer is: reflection to Quadrant III, then a translation of left parenthesis negative 1 comma 6 right parenthesis.