In a convex quadrilateral $ABCD$, it is given that $E$ is the intersection of the diagonals, triangles $ADE, BCE, CDE$ have areas of $12 \, \text{cm}^2, 45 \, \text{cm}^2, 18 \, \text{cm}^2$ respectively, and the length of side $AB$ is $7 \, \text{cm}$.

Determine the distance from point $D$ to the line $AB$.

(M. Petrová)
From the knowledge of the areas of the smaller triangles, we can determine the area of triangle $A B D$. From the knowledge of the length of side $A B$, we can determine the distance of point $D$ from line $A B$.

![](https://cdn.mathpix.com/cropped/2024_04_17_5c762d128fb89da6abfdg-5.jpg?height=423&width=1100&top_left_y=1913&top_left_x=478)

Triangles $A D E$ and $C D E$ share the side $D E$ and sides $A E$ and $E C$ lie on the same line. Therefore, the ratio of their areas is the same as the ratio of the lengths of sides $A E$ and $E C$,

$$
|A E|:|E C|=12: 18=2: 3 \text {. }
$$

Similarly, triangles $A B E$ and $B C E$ share the side $B E$ and sides $A E$ and $E C$ lie on the same line. The ratio of their areas is the same as the ratio of the lengths of sides $A E$ and $E C$,

$$
S_{A B E}: 45=2: 3 .
$$

From this, we get $S_{A B E}=30 \mathrm{~cm}^{2}$. The area of triangle $A B D$ is the sum of the areas of triangles $A B E$ and $A D E$,

$$
S_{A B D}=30+12=42\left(\mathrm{~cm}^{2}\right) .
$$

This area is equal to half the product of the length of side $A B$ and the distance of point $D$ from this side, $S_{A B D}=\frac{1}{2}|A B| \cdot v$. We know the area of $A B D$ and the length of $A B$, so the desired distance is

$$
v=\frac{2 \cdot 42}{7}=12(\mathrm{~cm})
$$

Evaluation. 4 points for the area of triangle $A B D$; 2 points for the desired distance and the quality of the commentary.

Notes. The considerations and relationships from the first part of the given solution can be succinctly written as $S_{A D E}: S_{C D E}=S_{A B E}: S_{B C E}$, equivalently as $S_{B C E}: S_{C D E}=S_{A B E}: S_{A D E}$, where the only unknown is the area $S_{A B E}$.

It also holds that the ratio of the areas $S_{A B D}: S_{B C D}$ is the same as the ratio of the heights of the triangles with respect to the common side $B D$, and this is the same as the ratio of the segments $|A E|:|E C|$. Meanwhile, $S_{B C D}=S_{B C E}+S_{C D E}$ is known from the problem statement, and the ratio $|A E|:|E C|$ is derived in (*). From this, it is possible to express $S_{A B D}$.