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.

Question

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.

[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]

GPT-4o mini · Answer

To find the area of the shaded region in the convex quadrilateral $ABCD$ with area $180$, where each side is trisected, we start by denoting the trisected points on each side as follows:

Let $P$ and $Q$ be the trisection points on side $AB$ such that $AP = PQ = QB$.
Let $R$ and $S$ be the trisection points on side $BC$.
Let $T$ and $U$ be the trisection points on side $CD$.
Let $V$ and $W$ be the trisection points on side $DA$.

The shaded region is bounded by the vertices $A$, $Q$, $R$, $C$, $U$, and $V$.

Next, let's consider the total area of the quadrilateral $ABCD$, which is given as $180$.

To find the area of the shaded region, we first determine how the area of quadrilateral $ABCD$ is distributed among the various triangles formed by the trisection points.

The key observation is that each side of the quadrilateral $ABCD$ contributes equally to the area of the triangles formed around the trisectors. Since there are 3 segments on each side and each segment divides the triangles formed, we need to find the fractions of areas effectively being encompassed by the triangles around the trisection points.

Step 1: Calculate the area ratio

Each pair of trisection points (like $P$ and $Q$ on $AB$) divides the area into three equal parts. The triangles at each corner of the quadrilateral can be calculated as follows:

Triangles formed adjacent to the vertices (e.g., triangle $APQ$) are one-third the area of the corresponding triangles made with the vertices.
This can be similarly said for all corners $B$, $C$, and $D$.

Step 2: Compute areas

Consequently, the shaded region includes parts of the triangle $ARC$ and triangle $AUV$, while the other parts correspond to triangles $ABR$, $BCD$, etc.

The area of each of the triangles at the vertices $A$, $B$, $C$, and $D$ contributes to the area of $ABCD$:

The total area of the quadrilateral $ABCD$ can therefore be divided into the shaded region plus the unshaded triangles around each vertex. Given that each triangle's area is one-sixth as they get divided uniformly across each side leading to:

Area of Triangles: \[ \text{Total area covered by vertices’ triangles} = \frac{1}{3} \text{(for each segment)} \times 4 (\text{for all vertices}) = \frac{1}{2} \text{ area of } ABCD. \]

Step 3: Find the shaded area

Thus the area of the shaded region: \[ \text{Shaded area} = \text{Area of } ABCD - \text{(Area covering four triangles)} \] Calculating using the fraction highlighted:

Each triangular segment accounts for $\frac{1}{6}$ of the entire $180$ area.
Thus, total triangle areas around the shape would total four times this: \[ 4 \times \text{Area of one segment} = 4 \times \frac{1}{6} \times 180 = 120. \]

Finally: \[ \text{Shaded Area} = 180 - 120 = 60. \]

Thus, the area of the shaded region is: \[ \boxed{60}. \]