To find the length of line segment \( X'Y' \), we first need to find the coordinates of points \( X' \) and \( Y' \) after the translation of \( XY \).
Step 1: Identify the original coordinates of points \( X \) and \( Y \)
- Point \( X \) is at \( (6, 5) \)
- Point \( Y \) is at \( (6, -4) \)
Step 2: Apply the translation
The translation consists of moving down 5 units and left 1 unit.
For point \( X \):
- Moving down 5 units: \( 5 - 5 = 0 \)
- Moving left 1 unit: \( 6 - 1 = 5 \)
- New coordinates for point \( X' \): \( (5, 0) \)
For point \( Y \):
- Moving down 5 units: \( -4 - 5 = -9 \)
- Moving left 1 unit: \( 6 - 1 = 5 \)
- New coordinates for point \( Y' \): \( (5, -9) \)
Step 3: Calculate the length of line segment \( X'Y' \)
The length of a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( X' \) and \( Y' \):
- \( X' = (5, 0) \)
- \( Y' = (5, -9) \)
Calculating the differences:
- \( x_2 - x_1 = 5 - 5 = 0 \)
- \( y_2 - y_1 = -9 - 0 = -9 \)
Now, substitute into the length formula:
\[ \text{Length} = \sqrt{(0)^2 + (-9)^2} = \sqrt{0 + 81} = \sqrt{81} = 9 \]
Conclusion:
The length of line segment \( X'Y' \) is \( 9 \) units.
Therefore, the final answer is: \[ \boxed{9} \text{ units} \]