a quadrilateral is plotted on the coordinate plane. Each of its vertices undergoes the same transformation.Which rule below represents a transformation that does not preserve the length of the sides of the quadrilateral?

There are two main types of transformations:
a) Rigid Transformations
b) Non-Rigid Transformations

Rigid Transformations, as their name indicate, do not alter the size or shape. These include reflection, rotation and translation

Non-Rigid Transformations, do not preserve the size. The shape remains the same. This include Dilation. 
The question asks to find a rule which do not preserve the lengths of the sides of the Quadrilateral. So one thing is for sure. The quadrilateral has undergone non-rigid transformation i.e Dilation. Dilation involves both contraction and expansion. So the rule which will either expand or contract the given quadrilateral will be the right choice to the given question.