To determine which transformation would not result in the two quadrilaterals being congruent, it's important to recall that congruence means that two shapes have the same size and shape.
Let's analyze the options provided:
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A translation along side line segment \( \overline{AB} \) by a distance of twice \( \overline{AB} \): Translations preserve size and shape, so this would result in a congruent quadrilateral.
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A reflection across side line segment \( \overline{BC} \): Reflections also preserve size and shape, thus this transformation would produce a congruent quadrilateral.
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A rotation about point \( C \) by 90 degrees counterclockwise: Rotations maintain the size and shape, indicating that this transformation would yield a congruent quadrilateral.
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A dilation from point \( C \) by a factor of one-half: Dilations change the size of the shape, which means the resulting quadrilateral will not be congruent to the original quadrilateral.
Based on this analysis, the transformation that would not result in the two quadrilaterals being congruent is:
A dilation from point \( C \) by a factor of one-half.