In the trapezoid $V O D Y$, it is given that $V O$ is the longer base, the intersection of the diagonals $K$ divides the segment $V D$ in the ratio $3: 2$, and the area of triangle $K O V$ is $13.5 \mathrm{~cm}^{2}$.

Determine the area of the entire trapezoid.

(M. Petrová)

Hint. What can you say about the other triangles contained in the trapezoid?
Since $V O$ is the longer base of the trapezoid $V O D Y$, the point $K$ on the diagonal $V D$ is closer to vertex $D$.

![](https://cdn.mathpix.com/cropped/2024_04_17_174124d98c91f026d57bg-1.jpg?height=460&width=734&top_left_y=2123&top_left_x=661)

Triangles $K O V$ and $K D O$ have a common height from vertex $O$, and the lengths of sides $V K$ and $K D$ corresponding to this height are in the ratio $3: 2$. Therefore, the areas of these triangles are also in the same ratio, that is,

$$
S_{K D O}=\frac{2}{3} S_{K O V}
$$

Triangles $V O D$ and $V O Y$ have a common side $V O$ and the same height to this side, so they have the same areas. Triangle $K O V$ is a part of both of these triangles, so the areas of triangles $K D O$ and $K Y V$ are also the same,

$$
S_{K Y V}=S_{K D O}=\frac{2}{3} S_{K O V}
$$

Triangles $K Y V$ and $K D Y$ have a common height from vertex $Y$, and the corresponding sides $V K$ and $K D$ are in the ratio $3: 2$. Therefore, the areas of these triangles are also in the same ratio,

$$
S_{K D Y}=\frac{2}{3} S_{K Y V}=\frac{4}{9} S_{K O V}
$$

The area of the entire trapezoid is the sum of the areas of the four mentioned triangles, that is,

$$
\begin{aligned}
S_{V O D Y} & =S_{K O V}+S_{K D O}+S_{K Y V}+S_{K D Y}= \\
& =\left(1+\frac{2}{3}+\frac{2}{3}+\frac{4}{9}\right) S_{K O V}=\frac{25}{9} \cdot 13.5=37.5\left(\mathrm{~cm}^{2}\right)
\end{aligned}
$$

Note. When sequentially calculating the areas of the above-mentioned triangles, we get

$$
S_{K D O}=S_{K Y V}=9 \mathrm{~cm}^{2}, \quad S_{K D Y}=6 \mathrm{~cm}^{2} .
$$

The equality $S_{K Y V}=\frac{2}{3} S_{K O V}$, or $S_{K D Y}=\frac{4}{9} S_{K O V}$, can be justified directly using the similarity of triangles $K O V$ and $K Y D$ (the similarity coefficient is $3: 2$).