To find the distance between the translated line segments \( A'B' \) and \( X'Y' \), we first need to determine the original positions of the segments and then apply the translations.
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Original Segments:
- \( XY \) has endpoints \( X(-2, 3) \) and \( Y(1, 3) \).
- \( AB \) has endpoints \( A(1, 1) \) and \( B(-2, 1) \).
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Translation of Segments:
- Both segments are translated 2 units left and 3 units down.
For segment \( XY \):
- \( X' = (-2 - 2, 3 - 3) = (-4, 0) \)
- \( Y' = (1 - 2, 3 - 3) = (-1, 0) \)
So the translated segment \( X'Y' \) has endpoints \( X'(-4, 0) \) and \( Y'(-1, 0) \).
For segment \( AB \):
- \( A' = (1 - 2, 1 - 3) = (-1, -2) \)
- \( B' = (-2 - 2, 1 - 3) = (-4, -2) \)
So the translated segment \( A'B' \) has endpoints \( A'(-1, -2) \) and \( B'(-4, -2) \).
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Finding Distance Between the Two Line Segments: The distance between the two parallel lines can be calculated by finding the distance between a point on one line to the other line.
The y-coordinates of segment \( A'B' \) (both points are at \( y = -2 \)) and \( X'Y' \) (both points are at \( y = 0 \)) indicate that the distance between the segments is the difference in their y-values:
\[ \text{Distance} = |0 - (-2)| = |0 + 2| = 2 \text{ units.} \]
Thus, the distance between the line segments \( A'B' \) and \( X'Y' \) is \(\boxed{2}\) units.