Use the image to answer the question. .

Question

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. Three triangles are formed by joining three plotted points each. Two of the triangles are joined by connecting their vertices with rays. The coordinates of the plotted points for the triangle in quadrant 4 are left parenthesis 1 comma negative 3 right parenthesis, left parenthesis 3 comma negative 3 right parenthesis, and left parenthesis 1 comma negative 7 right parenthesis. The coordinates for the triangle in quadrant 2 are as follows left parenthesis negative 5 comma 3 right parenthesis, left parenthesis negative 1 comma 3 right parenthesis, and left parenthesis negative 5 comma 5 right parenthesis. The side joining the vertices at left parenthesis negative 5 comma 3 right parenthesis and left parenthesis negative 5 comma 5 right parenthesis is labeled as e. The side joining the vertices at left parenthesis negative 5 comma 5 right parenthesis and left parenthesis negative 1 comma 3 right parenthesis is labeled as g. The side joining the vertices at left parenthesis negative 5 comma 3 right parenthesis and left parenthesis negative 1 comma 3 right parenthesis is labeled as h. The coordinates of the plotted points for the triangle in quadrant 1 are left parenthesis 3 comma 1 right parenthesis, left parenthesis 7 comma 1 right parenthesis, and left parenthesis 3 comma 3 right parenthesis. Three rays join the complementary vertices of the triangles in quadrants 1 and 2. A ray connects point left parenthesis 3 comma 1 right parenthesis and the point left parenthesis negative 5 comma 3 right parenthesis. A ray connects point left parenthesis 7 comma 1 right parenthesis and the point left parenthesis negative 1 comma 3 right parenthesis. A ray connects point left parenthesis 3 comma 3 right parenthesis and the point left parenthesis negative 5 comma 5 right parenthesis. The arrow heads are on the vertices of the triangle in quadrant 2.
Describe the series of transformations that have occurred to move the triangle in Quadrant IV to the triangle in Quadrant II to show that the triangles are congruent.(1 point)
Responses

translation, then rotation
translation, then rotation

rotation, then reflection
rotation, then reflection

reflection, then translation
reflection, then translation

rotation, then translation

Answer 1

To determine the transformations that have occurred to move the triangle in Quadrant IV to the triangle in Quadrant II, we first need to analyze the positions of the triangles:

1. Triangle in Quadrant IV has vertices at (1, -3), (3, -3), and (1, -7).
2. Triangle in Quadrant II has vertices at (-5, 3), (-1, 3), and (-5, 5).

To translate the triangle from Quadrant IV (in the positive x-direction) to Quadrant II (in the negative x-direction) and at higher y-coordinates, you likely first need to reflect the triangle across the y-axis or another line, and then translate it.

Following this reasoning:

• Reflecting the triangle in Quadrant IV across the y-axis would change the coordinates:
  • (1, -3) → (-1, -3)
  • (3, -3) → (-3, -3)
  • (1, -7) → (-1, -7)

This reflection positions the triangle in Quadrant III.

• After the reflection, we then translate the reflected triangle upwards to reach the coordinates of the Quadrant II triangle:
  • Translating the points upwards would involve moving them to reach the y-coordinates of the triangle in Quadrant II.

Based on this analysis, the best fit for the sequence of transformations moving the triangle from Quadrant IV to Quadrant II is:

reflection, then translation.

Thus, the correct response is: reflection, then translation.