To find the coordinates of the image quadrilateral \(A'B'C'D'\) under a dilation with a scale factor of \(4\), we need to first establish the coordinates of the original quadrilateral \(ABCD\).
The vertices of quadrilateral \(ABCD\) are given as follows:
- \(A(0, 3)\)
- \(B(2, 2)\)
- \(C(3, 3)\)
- \(D(3, 0)\)
Since the dilation has a scale factor of \(4\), the coordinates of the dilated vertices \(A', B', C', D'\) can be calculated by multiplying the coordinates of each original vertex by the scale factor of \(4\):
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For point \(A\): \[ A' = (0 \cdot 4, 3 \cdot 4) = (0, 12) \]
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For point \(B\): \[ B' = (2 \cdot 4, 2 \cdot 4) = (8, 8) \]
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For point \(C\): \[ C' = (3 \cdot 4, 3 \cdot 4) = (12, 12) \]
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For point \(D\): \[ D' = (3 \cdot 4, 0 \cdot 4) = (12, 0) \]
Now we have the coordinates of the dilated quadrilateral \(A'B'C'D'\):
- \(A'(0, 12)\)
- \(B'(8, 8)\)
- \(C'(12, 12)\)
- \(D'(12, 0)\)
Next, we need to find the length of segment \(\overline{C'D'}\). The coordinates of points \(C'\) and \(D'\) are \(C'(12, 12)\) and \(D'(12, 0)\).
The length of segment \(\overline{C'D'}\) can be calculated using the distance formula, which is given by:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Plugging in the coordinates \(C'(12, 12)\) and \(D'(12, 0)\) into the formula:
\[ \text{Distance} = \sqrt{(12 - 12)^2 + (0 - 12)^2} = \sqrt{0^2 + (-12)^2} = \sqrt{0 + 144} = \sqrt{144} = 12 \]
Thus, the length of segment \(\overline{C'D'}\) is \(\boxed{12}\).
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