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point W (2, 9)

point X (0, 9)
point Y (-1, 7)
point Z (3, 6)

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

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

a. ABCD was first rotated 90 (counterclockwise), then shifted 3 units to the left and 5 units up
b. ABCD was first shifted 3 units to the left and 5 units up, then rotated -90 (90 clockwise)
c. ABCD was first shifted 3 units to the left and 5 units up, then rotated -90 (90 clockwise)
d. ABCD was first rotated -270 (270 clockwise), then shifted 3 units to the left and 5 units up.

1 answer

To determine which two rigid transformations were performed on figure ABCD to produce the congruent figure WXYZ, we can analyze the coordinates of the points before and after the transformations.

  1. Current Coordinates:

    • Points A (-3, -2), B (-6, -5), C (-4, -6), D (-3, -2)
    • Points W (2, 9), X (0, 9), Y (-1, 7), Z (3, 6)

  2. Transformation Steps:

    • Step 1: Translating the points in ABCD. If we translate ABCD 3 units to the left and 5 units up, we change the coordinates as follows:
      • A (-3, -2) → A' (-3 - 3, -2 + 5) → A' (-6, 3)
      • B (-6, -5) → B' (-6 - 3, -5 + 5) → B' (-9, 0)
      • C (-4, -6) → C' (-4 - 3, -6 + 5) → C' (-7, -1)
      • D (-3, -2) → D' (-3 - 3, -2 + 5) → D' (-6, 3)

    Now the new points do not directly correspond to WXYZ, so we can try a different transformation first.

  3. Step 2: Rotating ABCD.

    • When rotating 90 degrees counterclockwise (or -270 degrees clockwise), the point transformation is:
      • A(x, y) → A'(-y, x)
      • B(-6, -5) → B'(5, -6)
      • C(-4, -6) → C'(6, -4)
      • D(-3, -2) → D'(2, -3)

    Translating these points:

    • A' (-2, -3) will result in ABCD being in the position where it might match WXYZ after appropriate translation.

  4. Finding the correct transformations: By testing the options:

  • Option a: Not applicable as rotation followed by shifting doesn't match.
  • Option b: Incorrect sequence; check against coordinates after translations.
  • Option c: Correctly indicates the operations.

If we apply the sequence: translating slightly incorrectly leads to matched points.

  1. Conclusion: After analyzing all transformations and keeping track of the coordinates through rotation and translation:

The correct answer will be b. ABCD was first shifted 3 units to the left and 5 units up, then rotated -90 (90 clockwise) as a correct directional transformation and alignment to yield corresponding points.