To find the coordinates of the new vertices after dilating triangle XYZ with a center at point \( C(4, 7) \) and a magnitude of \( \frac{1}{2} \), you can use the formula for dilation:
\[ X' = C + k(X - C) \]
where \( k \) is the scale factor (in this case, \( k = \frac{1}{2} \)), \( X \) is the original point, and \( C \) is the center of dilation.
The original vertices are:
- \( X(0, 4) \)
- \( Y(2, 0) \)
- \( Z(4, 7) \)
Calculating the new coordinates:
- For vertex \( X(0, 4) \):
\[ X' = (4, 7) + \frac{1}{2}((0, 4) - (4, 7)) \] \[ = (4, 7) + \frac{1}{2}((-4, -3)) \] \[ = (4, 7) + (-2, -1.5) \] \[ = (4 - 2, 7 - 1.5) = (2, 5.5) \]
- For vertex \( Y(2, 0) \):
\[ Y' = (4, 7) + \frac{1}{2}((2, 0) - (4, 7)) \] \[ = (4, 7) + \frac{1}{2}((-2, -7)) \] \[ = (4, 7) + (-1, -3.5) \] \[ = (4 - 1, 7 - 3.5) = (3, 3.5) \]
- For vertex \( Z(4, 7) \):
Since \( Z \) is the center of dilation, its coordinates will remain the same:
\[ Z' = Z(4, 7) \]
So, after the dilation, the coordinates of triangle \( X'Y'Z' \) are:
- \( X'(2, 5.5) \)
- \( Y'(3, 3.5) \)
- \( Z'(4, 7) \)
In summary, the new vertices after dilation are:
- \( X'(2, 5.5) \)
- \( Y'(3, 3.5) \)
- \( Z'(4, 7) \)
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