To find the coordinates of the dilated triangle \( X'Y'Z' \) from triangle \( XYZ \) using a dilation with a magnitude of 2 and a center of dilation at (2,0), we can use the formula for dilation:
\[ X' = C + k \cdot (X - C) \]
where \( C \) is the center of dilation, \( k \) is the magnitude of dilation, and \( X \) represents the original coordinate.
Given:
- Center of dilation \( C(2, 0) \)
- Magnitude \( k = 2 \)
For each vertex of triangle \( XYZ \), we apply the dilation formula:
- Let’s denote the coordinates of \( X, Y, \) and \( Z \) as \( (x, y) \).
The formula for the new coordinates will be:
\[ X' = \left(2 + 2(x - 2), 0 + 2(y - 0)\right) \] \[ = (2 + 2x - 4, 2y) \] \[ = (2x - 2, 2y) \]
This formula will be used for each original point \( (x, y) \) to derive the corresponding dilated points.
Let's analyze each option provided in terms of these calculations:
-
Option 1:
- \( X'(2,2), Y'(4,0), Z'(4.5,1) \)
- Check if they correspond to the dilation formula.
-
Option 2:
- \( X'(6,4), Y'(10,0), Z'(11,4) \)
- Check if they correspond to the dilation formula.
-
Option 3:
- \( X'(2,8), Y'(10,0), Z'(12,4) \)
- Check if they correspond to the dilation formula.
Since the original coordinates of triangle \( XYZ \) are not given in the problem statement directly, confirming the exact coordinates resulting from the dilation involves back-calculating from the listed options to match against the dilation criteria.
However, by noting that coordinates of \( X', Y', Z' \) should be derived from the stated relations and center point (2, 0) with a magnitude of 2, you can individually handle each of these responses to determine if they fit the derived pattern.
By exemplary calculation for one coordinated pair manually:
- If we assume an original vertex was \( (x,y)= (2,1) \) \[ X' = (2 \cdot 2 - 2, 2 \cdot 1) = (2,2) \]
- Dilation on others can yield us:
After processing options, truly providing original coordinates of triangle vertices is critical to identifying its match with the correct answers.
Overall, using the consistent application of the formula, check if any provided \( (x', y') \) reflect values derived with \( C(2,0) \) and the scaling factor to identify the option correctly.
The best approach amidst a lack of original data amid responses would deduce clarity through candidates providing valid results through calculated checks aligning with transformation rules as prompted in mathematics.