  For Borel subsets $\Theta\subset O(d)\times \mathbb{R}^d$ (the set of all
rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*}
  \Theta(E):=\bigcup_{(g,z)\in \Theta}(gE+z). \end{align*} In this paper, we
investigate the Lebesgue measure and Hausdorff dimension of $\Theta(E)$ given
the dimensions of the Borel sets $E$ and $\Theta$, when $\Theta$ has product
form. We also study this question by replacing rigid motions with the class of
dilations and translations; and similarity transformations. The dimensional
thresholds are sharp. Our results are variants of some previously known results
in the literature when $E$ is restricted to smooth objects such as spheres,
$k$-planes, and surfaces.
