To find the sequence of transformations that will take triangle \( \Delta ABC \) to triangle \( \Delta A"B"C" \), we need to compare the coordinates of the vertices in both triangles, applying transformations systematically.
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Coordinates of vertices:
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Triangle \( ABC \):
- A: (-5, 2)
- B: (-2, 4)
- C: (-3, 0)
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Triangle \( A"B"C" \):
- A": (-1, -2)
- B": (2, -4)
- C": (1, 0)
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Finding the transformations:
- First, let's try to analyze the transformations one step at a time starting from the original triangle's vertices to see how they can match the coordinates of triangle \( A"B"C" \).
Choice D: (x,y) → (x−4,−y)
Applying D to point A: \[ A: (-5, 2) \rightarrow (-5-4, -2) = (-9, -2) \] This point does not correspond to \(A"\).
Applying D to point B: \[ B: (-2, 4) \rightarrow (-2-4, -4) = (-6, -4) \] This point does not correspond to \(B"\).
Applying D to point C: \[ C: (-3, 0) \rightarrow (-3-4, 0) = (-7, 0) \] This point does not correspond to \(C"\).
We can eliminate transformation D.
Choice A: (x,y) → (x,−y+4)
Apply to point A: \[ A: (-5, 2) \rightarrow (-5, -2 + 4) = (-5, 2) \] A remains unchanged.
Apply A to point B: \[ B: (-2, 4) \rightarrow (-2, -4 + 4) = (-2, 0) \] This point does not correspond to \(B"\).
A does not work either.
Now, let's consider:
Choice B: (x,y) → (x+4,−y)
Apply to point A: \[ A: (-5, 2) \rightarrow (-5 + 4, -2) = (-1, -2) \] This corresponds to \( A" \).
Apply B to B: \[ B: (-2, 4) \rightarrow (-2 + 4, -4) = (2, -4) \] This corresponds to \( B" \).
Apply B to C: \[ C: (-3, 0) \rightarrow (-3 + 4, 0) = (1, 0) \] This corresponds to \( C" \).
Thus, a single transformation B successfully carries triangle \( ABC \) to \( A"B"C" \).
The final transformation mapping is: \[ \Delta ABC \xrightarrow{ (x,y) \rightarrow (x + 4, -y) } \Delta A"B"C" \]
So, the answer is: \[ (x,y) \rightarrow (x + 4, -y) \] (Choice B).
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