Apply coordinate rules to translate, reflect across the coordinate axes or horizontal and vertical lines, and rotate figures about the origin by $90^\circ$, $180^\circ$, or $270^\circ$, recognizing that rigid transformations preserve lengths, angle measures, shape, and congruence. Determine positive dilation scale factors from corresponding lengths or coordinates and dilate figures centered at the origin using $(x,y)\mapsto(kx,ky)$, distinguishing enlargements and reductions from rigid transformations. Perform ordered sequences of translations, reflections, rotations, and dilations to track figures and establish congruence through rigid motions or similarity through rigid motions followed by a dilation.