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

rotation, then reflection
rotation, then reflection

rotation, then translation
rotation, then translation

translation, then rotation
translation, then rotation

reflection, then translation
reflection, then translation
Skip to navigation

Answer 1

To determine the transformations that move the triangle in Quadrant IV to the triangle in Quadrant II, we need to analyze the positions of the triangles.

1. Coordinates of the triangle in Quadrant IV:

   • \( (1, -3) \)
   • \( (3, -3) \)
   • \( (1, -7) \)

2. Coordinates of the triangle in Quadrant II:

   • \( (-5, 3) \)
   • \( (-1, 3) \)
   • \( (-5, 5) \)

Next, we can look at how the points of Quadrant IV can be transformed into the points in Quadrant II.

1. Translation:

   • First, we can translate the triangle in Quadrant IV up to the x-axis (and beyond).
   • The translation of the y-coordinates from -3 to +3 requires an upward movement of 6 units.
   • To move from \( (1, -3) \) to \( (-5, 3) \), we also shift 6 units upward and 6 units leftward.

2. Reflection:

   • After we have moved the triangle upward, we can then reflect it across the y-axis.
   • This reflection takes the transformed coordinates from the translation and flips them to the opposite side of the y-axis.

From this analysis, the triangles are congruent through a combination of translation followed by reflection.

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