An equilateral triangle of mass \(M\) and side-length \(L\) is cut from uniformdensity sheet metal.

(a) Draw the triangle along with the three perpendicular bisectors, each of which extends from the middle of a side to the opposite vertex. Show that each bisector has length \(\sqrt{3} L / 2\).

(b) Explain why the center of mass of the triangle must be located at the point where the three perpendicular bisectors intersect. Let this point be the origin.

(c) Let the \(z\) axis be perpendicular to the triangle, the \(y\) axis be along one of the perpendicular bisectors, and the \(x\) axis be perpendicular to both. Find the moment of inertia matrix for the triangle in these coordinates.