Identify a sequence of rigid transformations that would map

ABCD→A"B"C"D" . Give your answer in the form of a composition transformation mapping.

A. (x,y)→(x,−y−4)
B. (x,y)→(−x−4,y)
C. (x,y)→(x,y+4)
D.(x,y)→(−x,y−4)

Rectangle ABCD
Point A: (-4, 3)
Point B: (-2, 4)
Point C: (-1, 1)
Point D: (-3, 0)

Rectangle A"B"C"D"
Point A": (4, -1)
Point B": (2, 0)
Point C": (1, -3)
Point D": (3, -4)

1 answer

To find the sequence of rigid transformations (which include translations, rotations, and reflections) that maps the rectangle ABCD to A"B"C"D", we will analyze the transformations one by one.

  1. Identify the changes in coordinates:

    • Point A: \( A(-4, 3) \) transforms to \( A"(4, -1) \).
    • Point B: \( B(-2, 4) \) transforms to \( B"(2, 0) \).
    • Point C: \( C(-1, 1) \) transforms to \( C"(1, -3) \).
    • Point D: \( D(-3, 0) \) transforms to \( D"(3, -4) \).
  2. Determine the transformations needed:

    • The x-coordinates change from negative to positive, indicating a reflection over the y-axis.
    • The y-coordinates decrease as the x-coordinates increase, suggesting a vertical shift downward.
  3. Testing the transformations: Let's analyze the given options to see which fit:

  • Option A: \( (x,y) \to (x, -y-4) \)

    • A (-4, 3) → (-4, -3-4) = (-4, -7) (not matching)
  • Option B: \( (x,y) \to (-x-4, y) \)

    • A (-4, 3) → (4-4, 3) = (0, 3) (not matching)
  • Option C: \( (x,y) \to (x, y+4) \)

    • A (-4, 3) → (-4, 3+4) = (-4, 7) (not matching)
  • Option D: \( (x,y) \to (-x, y - 4) \)

    • This performs a reflection across the y-axis and then moves down by 4 units. \
    • A (-4, 3) → (4, 3-4) = (4, -1) → A"](4, -1) (this matches!)

Let's confirm for all points with Option D:

  • B (-2, 4) → (2, 4 - 4) = (2, 0) → B"](2, 0) (this matches!)
  • C (-1, 1) → (1, 1 - 4) = (1, -3) → C"](1, -3) (this matches!)
  • D (-3, 0) → (3, 0 - 4) = (3, -4) → D"](3, -4) (this matches!)

Conclusion:

The transformation that maps rectangle ABCD to rectangle A"B"C"D" is: Option D: (x, y) → (-x, y - 4)

Thus, the composition of transformations is:

  • Reflect over the y-axis, and
  • Translate downward by 4 units.

In formal notation, it is: \[ D \circ (x,y) \to (-x, y - 4) \]