To find the new coordinates of the vertices \( X' \), \( Y' \), and \( Z' \) after dilating triangle \( XYZ \) with a center of dilation at \( (4, 7) \) and a magnitude of \( \frac{1}{2} \), we use the formula for dilation:
\[ X' = (1 - k)C + kX \]
where \( k \) is the dilation factor, \( C \) is the center of dilation, and \( X \) is the original point.
Given:
- Center of dilation \( C = (4, 7) \)
- Dilation factor \( k = \frac{1}{2} \)
Step 1: Calculate the new coordinates for vertex \( X(0, 4) \)
\[ X' = \left( 4 + \frac{1}{2}(0 - 4), 7 + \frac{1}{2}(4 - 7) \right) \] \[ X' = \left( 4 - 2, 7 - \frac{3}{2} \right) \] \[ X' = (2, 5.5) \]
Step 2: Calculate the new coordinates for vertex \( Y(2, 0) \)
\[ Y' = \left( 4 + \frac{1}{2}(2 - 4), 7 + \frac{1}{2}(0 - 7) \right) \] \[ Y' = \left( 4 - 1, 7 - \frac{7}{2} \right) \] \[ Y' = (3, 3.5) \]
Step 3: Calculate the new coordinates for vertex \( Z(4, 7) \)
Since \( Z \) is the center of dilation: \[ Z' = (4, 7) \quad \text{(remains the same)} \]
Final Result
Thus, the coordinates of the new vertices \( X' \), \( Y' \), and \( Z' \) after dilation are: \[ X' = (2, 5.5), \quad Y' = (3, 3.5), \quad Z' = (4, 7) \]
The correct answer is:
D. \( X'(2, 5.5), Y'(3, 3.5), Z'(4, 7) \).