Identify a sequence of rigid transformations that would map

Question

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

(1 point)

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)

Answer 1

To find the sequence of rigid transformations that maps rectangle ABCD to rectangle A"B"C"D", we can analyze the position and orientation of the points provided.

1. Identify the transformation that moves the rectangle vertically downwards. We notice the y-coordinates of A, B, C, and D decrease from A: 3 to A": -1, which indicates a downward transformation.

   • The transformation \( (x,y) \to (x,y-4) \) would move point A to ( -4, 3 - 4) = (-4, -1), which is close but not the exact point A".
     However, the vertical transformation should indeed lower the figure.

2. Identify the transformation that reflects the rectangle. Additionally, we observe that the x-coordinates should flip and also need to shift.

   • First, transforming by \( (x, y) \to (-x, y-4) \) will move A to: (4, -1).
   • When applying \( (x,y) \to (-x,y-4) \) to the point A: (-4,3):
     • A: (-4,3) → A": (4,3-4) = (4,-1)

3. Conclusion - combination of transformations. Thus, the following composition maps ABCD to A"B"C"D":

• Apply Reflection (B): Apply \( (x,y) \to (-x-4,y) \):
  \( A(-4,3) \rightarrow (4,3)\)
  \( B(-2,4) \rightarrow (2,4)\)
  \( C(-1,1) \rightarrow (1,1)\)
  \( D(-3,0) \rightarrow (3,0)\)

This shifts all in the x-direction.

• Now apply downward shift (C or D): For our purpose: \( (x,y) \to (x,y-4) \):
  So \( A"(4,3)\) becomes \( A"(4,-1)\)
  \( B"(2,4) \rightarrow (2,0)\)
  \( C"(1,1) \rightarrow (1,-3)\)
  \( D"(3,0) \rightarrow (3,-4)\)

Final Transformation

The full transformation to swap from ABCD to A"B"C"D": The final option should be the composition of these transformations:

1. \( (x,y) \rightarrow (-x-4,y) \) (reflect)
2. \( (x,y) \rightarrow (x, y-4) \) (shift down)

Therefore the complete transformation can be described as: \[ (x,y) \to (-x-4,y) \] followed by \[ (x,y) \to (x,y-4) \]

So the answer is: B followed by D In notation: \( D \circ B \).

Final Answer

(B, then D): \[ ((x,y) \to (-x-4,y)) \text{ followed by } ((x,y) \to (x,y-4)) \]