Answer:
When a figure is dilated with a scale factor \( k \) from a center point \( (h, k) \), the coordinates of the new point \( (x', y') \) are given by:
\[ x' = h + k \cdot (x - h) \]
\[ y' = k \cdot (y - k) \]
In this case, the center of dilation is \( (1, 1) \) and the scale factor is \( \frac{1}{2} \). Let's assume the coordinates of point B are \( (x_B, y_B) \) and the coordinates of B' are \( (x_{B'}, y_{B'}) \).
\[ x_{B'} = 1 + \frac{1}{2} \cdot (x_B - 1) \]
\[ y_{B'} = \frac{1}{2} \cdot (y_B - 1) \]
Now, to find the length of B'C', you need to calculate the distance between B and C in the original quadrilateral ABCD and then apply the scale factor:
\[ \text{Length of B'C'} = \frac{1}{2} \times \text{Length of BC} \]
If you have the coordinates of B and C, you can use the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Repeat this process for each pair of corresponding points in the original and dilated quadrilaterals (B and B', C and C').
Keep in mind that this assumes a dilation without a change in orientation. If there's a rotation involved, the calculations would be more complex.