Moment in parallelogram

Question

If F is a force in the same plane as parallelogram ABCD and the moment of F about A equals -18 moment unit and the moment about B equals the moment about D equals 32, what is the moment of F about C? The answer is 82 but I want to know how

Answer 1

Consider a number $n$ of points in the plane and the relevant centroid (= barycenter, for points with the same mass).

Take any oriented line passing through the barycenter: by definition, the sum of the distances (with sign) from each point to the line is null. Same will be the moment of a force placed on that line.

The sum of the moments for a force parallel to that line will be $n$ times its moment wrt the barycenter.

If the points, like in your case, are arranged by couples symmetric to the (same) barycenter, then each couple will provide the same sum of moments, i.e. two times the moment of the force wrt the barycenter.

In your case you are given the couple BD which sum to $64$. Same shall be the sum of the moments for the couple AC: $82-18=64$