Congruent Figures Quick Check

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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

reflection, then translation
reflection, then translation

translation, then rotation
translation, then rotation

rotation, then reflection
rotation, then reflection

rotation, then translation
rotation, then translation

Answer 1

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

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

Transformation Steps:

• Reflection: The triangle in Quadrant IV can be reflected over the y-axis. This moves the points (1, -3) to (-1, -3) and (3, -3) to (-3, -3), while point (1, -7) moves to (-1, -7). Reflecting in the y-axis causes the x-coordinates to change sign.

• Translation: After reflecting, we now translate the triangle upward to match the y-coordinates of the triangle in Quadrant II. This involves adding a certain distance to the y-coordinates, specifically from y = -3 or -7 to y = 3.

Thus, the transformation sequence involves starting with a reflection over the y-axis followed by a translation upward to match the positions of the second triangle.

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