+8 In the diagram below, each side of convex quadrilateral is trisected. (For example, ) The area of convex quadrilateral is 180. Find the area of the shaded region.
[asy] unitsize(1 cm); pair A, B, C, D, P, Q, R, S, T, U, V, W; A = (1,2); B = (4,3); C = (5,-1); D = (0,0); P = (2*A + B)/3; Q = (A + 2*B)/3; R = (2*B + C)/3; S = (B + 2*C)/3; T = (2*C + D)/3; U = (C + 2*D)/3; V = (2*D + A)/3; W = (D + 2*A)/3; fill(A--Q--R--C--U--V--cycle,gray(0.7)); draw(A--B--C--D--cycle); draw(Q--R); draw(U--V); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, SE); dot("$D$", D, SW); dot("$P$", P, N); dot("$Q$", Q, N); dot("$R$", R, E); dot("$S$", S, E); dot("$T$", T, dir(270)); dot("$U$", U, dir(270)); dot("$V$", V, NW); dot("$W$", W, NW); [/asy]
+544 Let's denote the area of the shaded region as S.
Observation: Notice that the shaded region consists of 4 smaller quadrilaterals: APWV, PQRS, RTUC, and UVWD. Each of these smaller quadrilaterals is similar to the original quadrilateral ABCD.
Ratio of Areas: Since the sides of the smaller quadrilaterals are one-third the length of the corresponding sides of ABCD, their areas will be one-ninth the area of ABCD.
Calculating the Area of the Shaded Region: Therefore, the area of each smaller quadrilateral is 180/9=20.
So, the total area of the shaded region, S, is:
S=4∗20=80
Therefore, the area of the shaded region is 80.
