To find the coordinates of the new vertices \( X' \), \( Y' \), and \( Z' \) after dilating triangle XYZ with a center of dilation at \( (4,7) \) and a magnitude of \( 12 \), we apply the formula for dilation:
\[ X' = \left( (1 - k) \cdot x + k \cdot x_c, (1 - k) \cdot y + k \cdot y_c \right) \]
Where:
- \( (x, y) \) are the coordinates of the original vertex,
- \( (x_c, y_c) \) are the coordinates of the center of dilation,
- \( k \) is the dilation factor (in this case, \( k = 12 \)).
Step 1: Calculate the scale factor \( k \)
The scale factor for dilation, when given as a magnitude, needs to be converted to a ratio using the distance to the center. However, for dilation, the distances just multiply by the scale factor.
Step 2: Calculate for point \( X(0, 4) \)
\[ X' = \left( (1 - 12) \cdot 0 + 12 \cdot 4, (1 - 12) \cdot 4 + 12 \cdot 7 \right) \]
Calculating each coordinate:
- For the x-coordinate: \[ X'_x = 12 \cdot (0 - 4) + 4 = -48 + 4 = -44 \]
- For the y-coordinate: \[ X'_y = 12 \cdot (4 - 7) + 7 = 12(-3) + 7 = -36 + 7 = -29 \]
Thus, \( X'(-44, -29) \).
Step 3: Calculate for point \( Y(2, 0) \)
\[ Y' = \left( (1-12) \cdot 2 + 12 \cdot 4, (1 - 12) \cdot 0 + 12 \cdot 7 \right) \]
Calculating each coordinate:
- For the x-coordinate: \[ Y'_x = 12 \cdot (2-4) + 4 = -24 + 4 = -20 \]
- For the y-coordinate: \[ Y'_y = 12 \cdot (0 - 7) + 7 = -84 + 7 = -77 \]
Thus, \( Y'(-20, -77) \).
Step 4: Calculate for point \( Z(4, 7) \)
Since point \( Z \) is the center of dilation: \[ Z' = (4,7) \]
Combined Result
So the new vertices after dilation are:
- \( X'(-44, -29) \)
- \( Y'(-20, -77) \)
- \( Z'(4, 7) \)
Comparing against the answer choices given in your question, it appears that none of the provided options correspond to the calculations performed.
If we assume that the dilation is uniform but there might be a typo or misunderstanding in the magnitude provided or its representation, please let me know to clarify or explore further adjustments.