New subscriber, please help:stuck on this solid body mechanics quiz question

I haven't been in school for a while, but the first thing that comes to my mind is the centroid of area. See this link:

I suppose the quiz is not related to an assessment.

The principal axes have zero sectional product of inertia, and they are perpendicular to each other.

If the section is symmmetric, the line of symmetry is one of the priincipal axes.

John

To put it in slightly different terms: You'll need to find the centroid of the cross section and then find the eigenvalues/vectors of the inertia matrix about the centroid. The tough part is figuring out what detail of that your professor has singled out as the "key characteristic". That's your call.