To find the coordinates of the new vertices \(X′Y′Z′\) after dilating triangle \(XYZ\) with a center of dilation at \(C(4,7)\) and a magnitude of \( \frac{1}{2} \), we can use the formula for dilation.
The formula for dilation of a point \( (x, y) \) from a center of dilation \( (x_c, y_c) \) is given by: \[ (x', y') = (x_c + k(x - x_c), y_c + k(y - y_c)) \] where \( k \) is the scale factor.
Now, let’s find the new coordinates of each vertex:
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Vertex X(0, 4): \[ X' = (4 + \frac{1}{2}(0 - 4), 7 + \frac{1}{2}(4 - 7)) \] \[ X' = (4 - 2, 7 - 1.5) = (2, 5.5) \]
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Vertex Y(2, 0): \[ Y' = (4 + \frac{1}{2}(2 - 4), 7 + \frac{1}{2}(0 - 7)) \] \[ Y' = (4 - 1, 7 - 3.5) = (3, 3.5) \]
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Vertex Z(4, 7): \[ Z' = (4 + \frac{1}{2}(4 - 4), 7 + \frac{1}{2}(7 - 7)) \] \[ Z' = (4 + 0, 7 + 0) = (4, 7) \]
Thus, the new vertices after dilation are:
- \(X′(2, 5.5)\)
- \(Y′(3, 3.5)\)
- \(Z′(4, 7)\)
The correct response from the options provided is: X′(2,5.5), Y′(3,3.5), Z′(4,7).