How did the authors determine both the spatial size of gas cloud HCN-0.009-0.044 and its central mass at the same time? Phys.org's Hiding black hole found says:

A research team led by Shunya Takekawa at the National Astronomical Observatory of Japan noticed HCN-0.009-0.044, a gas cloud moving strangely near the center of the galaxy 25,000 light-years away from Earth in the constellation Sagittarius. They used ALMA (Atacama Large Millimeter/submillimeter Array) to perform high-resolution observations of the cloud and found that it is swirling around a massive invisible object.
Takekawa says, &quot;Detailed kinematic analyses revealed that an enormous mass, 30,000 times that of the sun, was concentrated in a region much smaller than our solar system. This and the lack of any observed object at that location strongly suggests an intermediate-mass black hole. By analyzing other anomalous clouds, we hope to expose other quiet black holes.&quot;
Tomoharu Oka, a professor at Keio University and coleader of the team, adds, &quot;It is significant that this intermediate mass black hole was found only 20 light-years from the supermassive black hole at the galactic center. In the future, it will fall into the supermassive black hole, much like gas is currently falling into it. This supports the merger model of black hole growth.&quot;
These results were published as Takekawa et al. &quot;Indication of Another Intermediate-mass Black Hole in the Galactic Center&quot; in The Astrophysical Journal Letters on January 20, 2019.

The paper Indication of Another Intermediate-mass Black Hole in the Galactic Center (open access) is pretty hard to read as it details the careful analysis of ALMA data reduction and analysis.
Question: How did the authors determine both the spatial size of gas cloud HCN-0.009-0.044 and the mass of the central object at the same time?
The gas clouds are much larger than you think: From Fig. 1 of the paper (Takakawa 2019), the largest cloud (which they call the "Balloon") is roughly 10 arcsec across. Now they can't actually measure its exact distance, but its projected distance from the Galactic center is only 7 pc, and the authors argue that it's at least 5 kpc away due to absorption from molecular gas which is known at this distance (Sofue 2006), and since such hot and high density ($\sim 10^7\,\mathrm{cm}^{-3}$) molecular gas is abundant in the Galactic center, it is probably located there, i.e. at a distance of $R=8\,\mathrm{kpc}$.

The cloud's absolute size is thus
$$
D = 10'' \times 8\,\mathrm{kpc} = 80\,000\,\mathrm{AU} \simeq 0.4\,\mathrm{pc}.
$$

The mass is measured from dynamics, i.e. they measure the line-of-sight velocities of several clouds. These velocities depend both on the mass of the dynamical center (i.e. whatever object the clouds orbit), and on the total velocity of this object wrt. us. Since they only know the line-of-sight velocities and not the full, 3D motions, they cannot know with certainty the mass, but performing a chi-square fit of 3D models that fit the 2D observations, they determine the best-fit value of the mass to be $(3.2\pm0.6)\times10^4 M_\odot$.

Their Fig. 4 shows the model:



3D model of the motion of the clouds. The red cloud is the Balloon. Other observed clouds are shown in green, blue, and orange colors. The black dot indicates the position of the black hole. The actual observation is the 2D projection, seen in the bottom plane.

The model of the velocities also gives the actual trajectories of the clouds around the central mass. Because the radius of the Balloon's orbit is $\sim0.07\,\mathrm{pc}$, it means that this much mass must be concentrated within a region of maximum that size, and probably significantly smaller. This implies a density of $&gt;2\times10^8 M_\odot\,\mathrm{pc}^{-3}$, at least an order of magnitude larger than even the most densely packed globular clusters, and since no luminous counterpart is observed at that location, the authors consider an intermediate-mass black hole as the most promising candidate for the gravitational source.