In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex quadrilateral $ABCD$ is $180.$ Find the area of the shaded region.
[ABCD] = AD * DC = 180
The shaded area is a trapezoid with area = (1/2) AD (AB + UT) and AB = DC and UT = DC / 3
So....the shaded area = (1/2)(AD) (DC + DC/3) = (1/2)(AD)(DC) + (1/2)(AD)(DC//3) =
90 + (1/6)(AD)(DC) =
90 + 30 =
120
+1911 
