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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 10 to 10 in increments of 1. The y axis ranges from negative 10 to 10 in increments of 1.

Image Long DescriptionPoint upper A has coordinates left parenthesis negative 6 comma negative 3 right parenthesis. Point upper B has coordinates left parenthesis negative 6 comma negative 5 right parenthesis. Point upper C has coordinates left parenthesis negative 4 comma negative 6 right parenthesis. Point upper D has coordinates left parenthesis negative 3 comma negative 2 right parenthesis. Point upper W has coordinates left parenthesis 2 comma 9 right parenthesis. Point upper X has coordinates left parenthesis 0 comma 9 right parenthesis. Point upper Y has coordinates left parenthesis negative 1 comma 7 right parenthesis. Point upper Z has coordinates left parenthesis 3 comma 6 right parenthesis.

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

(1 point)
Responses

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°
(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 (90 degrees clockwise).

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°
(270°
clockwise), then shifted 3 units to the left and 5 units up.

1 answer

To determine the two rigid transformations performed on quadrilateral ABCD to produce the congruent figure WXYZ, we can analyze the coordinates of both figures.

  1. Coordinates of Quadrilateral ABCD:

    • A (-6, -3)
    • B (-6, -5)
    • C (-4, -6)
    • D (-3, -2)

  2. Coordinates of Quadrilateral WXYZ:

    • W (2, 9)
    • X (0, 9)
    • Y (-1, 7)
    • Z (3, 6)

Now, we will calculate the transformations:

Step 1: Shifting

  • From point A (-6, -3) to point W (2, 9):
    • The change in x-coordinates: 2 - (-6) = 2 + 6 = 8
    • The change in y-coordinates: 9 - (-3) = 9 + 3 = 12
  • This indicates a shift right (8 units) and up (12 units).

Step 2: Rotation

After shifting the points, we need to analyze if a rotation transforms the new coordinates of ABCD into the coordinates of WXYZ.

If we calculate how much we need to rotate to match the transformation, we find that rotating -90° (or 90° clockwise) could match the respective new positions.

Thus, we can conclude that the transformations are as follows:

  • First transformation: Shifted right 8 units and up 12 units (this matches with WXYZ).
  • Second transformation: A further rotation of -90° (90° clockwise) makes the points align correctly.

From the transformation options provided: ABCD was first shifted 3 units to the left and 5 units up, then rotated -90° (90° clockwise).

This is clearly not matching since we have already established it was shifted right? Hence, the transformations must thus align with the option given that it involved analysis derived earlier:

  • ABCD was first rotated 90° (counterclockwise), then shifted parts accordingly.

Correct response:

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