To find the vertices of triangle \( A'B'C' \) after the translation \( T_{(-4,5)}( \triangle ABC) \), we will apply the translation to each vertex of triangle \( ABC \).
The translation \( T_{(-4,5)} \) means that we will subtract 4 from the x-coordinate and add 5 to the y-coordinate of each vertex.
Here are the vertices of triangle \( ABC \):
- \( A(2, -1) \)
- \( B(-2, -3) \)
- \( C(4, 3) \)
Now we will translate each vertex:
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For Vertex A: \[ A' = (2 - 4, -1 + 5) = (-2, 4) \]
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For Vertex B: \[ B' = (-2 - 4, -3 + 5) = (-6, 2) \]
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For Vertex C: \[ C' = (4 - 4, 3 + 5) = (0, 8) \]
Now we can summarize the vertices of triangle \( A'B'C' \):
- \( A'(-2, 4) \)
- \( B'(-6, 2) \)
- \( C'(0, 8) \)
Therefore, the vertices of triangle \( A'B'C' \) are:
- \( A'(-2, 4) \)
- \( B'(-6, 2) \)
- \( C'(0, 8) \)
In conclusion:
- A' = (-2, 4)
- B' = (-6, 2)
- C' = (0, 8)