## Task 2 - 320732

Let $A B C D$ be a quadrilateral whose diagonals $A C$ and $B D$ intersect at a point $S$. Furthermore, it is assumed that the triangles $A B S, D A S$, and $B C S$ have areas of $1000 \mathrm{~cm}^{2}, 800 \mathrm{~cm}^{2}$, and $1200 \mathrm{~cm}^{2}$, respectively, as indicated in the figure.

Show that the area $F$ of the triangle $C D S$ is uniquely determined by these assumptions, and determine this area!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0540.jpg?height=394&width=440&top_left_y=1856&top_left_x=1322)
In the triangles $B C S$ and $D A S$, let $e$ and $f$ be the lengths of the sides $B S$ and $D S$, and $h$ and $k$ be the lengths of the heights perpendicular to these sides.

Then, $k$ and $h$ are also the lengths of the heights perpendicular to the sides $B S$ and $D S$ in the triangles $A B S$ and $C D S$. Therefore, the given areas and the desired area are

$$
\frac{e \cdot k}{2}=1000 \mathrm{~cm}^{2}, \quad \frac{f \cdot k}{2}=800 \mathrm{~cm}^{2}, \quad \frac{e \cdot h}{2}=1200 \mathrm{~cm}^{2}, \quad \frac{f \cdot h}{2}=F
$$

From this, it follows that

$$
f: e=800: 1000=4: 5, \quad f=\frac{4 e}{5}
$$

i.e., the conditions uniquely determine

$$
F=\frac{4 e \cdot h}{5 \cdot 2}=\frac{4}{5} \cdot 1200 \mathrm{~cm}^{2}=960 \mathrm{~cm}^{2}
$$