  We calculate root-mean-square radii for a three-body system confined to two
spatial dimensions and consisting of two identical bosons ($A$) and one
distinguishable particle ($B$). We use zero-range two-body interactions between
each of the pairs, and focus thereby directly on universal properties. We solve
the Faddeev equations in momentum space and express the mean-square radii in
terms of first-order derivatives of the Fourier transforms of densities. The
strengths of the interactions are adjusted for each set of masses to produce
equal two-body bound-state energies between different pairs. The mass ratio,
${\cal A}=m_B/m_A$, between particles $B$ and $A$ are varied from $0.01$ to
$100$ providing a number of bound states decreasing from $8$ to $2$. Energies
and mean-square radii of these states are analyzed for small ${\cal A}$ by use
of the Born-Oppenheimer potential between the two heavy $A$-particles. For
large ${\cal A}$ the radii of the two bound states are consistent with a
slightly asymmetric three-body structure. When ${\cal A}$ approaches thresholds
for binding of the three-body excited states, the corresponding mean-square
radii diverge inversely proportional to the deviation of the three-body energy
from the two-body thresholds. The structures at these three-body thresholds
correspond to bound $AB$-dimers and one loosely bound $A$-particle.
