To determine the scale factor of the dilation \( AC \) with respect to the center of dilation \( B \), we can use the formula for the scale factor \( k \) given by the ratio of the distances:
\[ k = \frac{\text{distance from } B \text{ to } C}{\text{distance from } B \text{ to } A} \]
First, we need to find the distances from point \( B \) to points \( A \) and \( C \).
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Calculate the distance from \( B \) to \( A \):
Point \( A = (1, -1) \) and point \( B = (2, 2) \).
\[ \text{Distance } d(B, A) = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] \[ d(B, A) = \sqrt{(2 - 1)^2 + (2 - (-1))^2} \] \[ d(B, A) = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Calculate the distance from \( B \) to \( C \):
Point \( C = (5, 5) \) and point \( B = (2, 2) \).
\[ d(B, C) = \sqrt{(x_B - x_C)^2 + (y_B - y_C)^2} \] \[ d(B, C) = \sqrt{(2 - 5)^2 + (2 - 5)^2} \] \[ d(B, C) = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Now calculate the scale factor \( k \):
\[ k = \frac{d(B, C)}{d(B, A)} = \frac{3\sqrt{2}}{\sqrt{10}} \]
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Simplify the scale factor:
To simplify, we can multiply the numerator and the denominator by \( \sqrt{10} \):
\[ k = \frac{3\sqrt{2}}{\sqrt{10}} = \frac{3\sqrt{2} \cdot \sqrt{10}}{10} = \frac{3\sqrt{20}}{10} = \frac{3 \cdot 2\sqrt{5}}{10} = \frac{6\sqrt{5}}{10} = \frac{3\sqrt{5}}{5} \]
Thus, the scale factor \( k \) of the dilation \( AC \) with respect to point \( B \) is:
\[ \boxed{\frac{3\sqrt{5}}{5}} \]
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