Final answer:
The transformation (x,y) —> (2x+1, y-5) does not preserve the lengths of the sides of a quadrilateral because it scales the x-coordinates, unlike other transformations like translations and reflections which are rigid motions.
Step-by-step explanation:
The transformation that does not preserve the lengths of the sides of a quadrilateral is (x,y) —> (2x+1, y-5). This transformation is not a rigid motion because it involves scaling the x-coordinates of the points, which changes the distances between points. On a coordinate plane, the length of a vector or side of a shape is preserved under movements like translations, rotations, and reflections because these are rigid motions that don't alter the size or shape of figures.
A translation transformation like (x,y) —> (x-3, y+2) simply shifts each point horizontally by 3 units to the left and vertically by 2 units up; it preserves lengths and angles. The transformation (x,y) —> (-x+6, -y+10) is a reflection across the origin followed by a translation, which also preserves lengths and angles. The last transformation, (x,y) —> (-x-5, y+10), is similarly a combination of reflection and translation, preserving side lengths. The transformation that fails to preserve lengths involves a change in scale (dilatation), specified by the 2x, which alters the distance between points.