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Two quadrilaterals, upper A upper B upper C upper D and upper W upper X upper Y upper Z, are graphed on a coordinate plane. The x-axis ranges from negative 14 to 10 in increments of 2. The y-axis ranges from negative 6 to 10 in increments of 2.

What two rigid transformations were performed on figure ABCD to produce the congruent figure WXYZ?

(1 point)
Responses

ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).
upper A upper B upper C upper D was first shifted 3 units to the left and 5 units up, then rotated 90 degrees (counterclockwise).

ABCD was first rotated −270° (clockwise), then shifted 3 units to the left and 5 units up.
upper A upper B upper C upper D was first rotated negative 270 degrees (clockwise), then shifted 3 units to the left and 5 units up.

ABCD was first rotated 90° (counterclockwise), then shifted 3 units to the left and 5 units up.
upper A upper B upper C upper D was first rotated 90 degrees (counterclockwise), then shifted 3 units to the left and 5 units up.

ABCD was first shifted 3 units to the left and 5 units up, then rotated −90° (clockwise).
upper A upper B upper C upper D was first shifted 3 units to the left and 5 units up, then rotated negative 90 degrees (clockwise).
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1 answer

To determine the rigid transformations performed on figure ABCD to produce figure WXYZ, we need to analyze the changes in both position and orientation of the quadrilaterals on the coordinate plane.

  1. Shifting (Translation): The transformation of shifting (or translating) a figure involves moving it a certain distance in a specific direction. In this case, shifting 3 units to the left and 5 units up means subtracting 3 from the x-coordinates and adding 5 to the y-coordinates of all vertices of figure ABCD.

  2. Rotation: After the translation, the figure can be rotated, which involves turning the figure around a fixed point (often the origin in this context). A 90° counterclockwise rotation would involve swapping the coordinates and changing the sign of the new x-coordinate (original y-coordinate) for the transformation.

To find the correct combination of transformations for the quadrilateral ABCD to match WXYZ:

  • Check the movement of each vertex after shifting and rotating.
  • Note the final orientation of WXYZ in comparison to ABCD.

Based on the options provided, the correct transformation sequence should:

  1. Shift ABCD left and up.
  2. Rotate the resulting figure to achieve the final orientation of WXYZ.

Among the options given: The choice that clearly outlines:

  • Shifting 3 units to the left and 5 units up, and
  • Then rotating 90° counterclockwise,

is:

ABCD was first shifted 3 units to the left and 5 units up, then rotated 90° (counterclockwise).

This sequence maintains the congruence between the two figures.