Problem 5. In trapezoid $ABCD$, diagonal $AC$ divides the midline into segments of lengths $a$ and $b$. Determine the ratio of the areas of triangles $\triangle ABC$ and $\triangle ACD$.
Solution. Let $h$ be the height of the trapezoid. The segments $a$ and $b$ of the midline of the trapezoid are midlines for triangles $\triangle A C D$ and $\triangle A B C$, respectively. Therefore, $\overline{A B}=2 b$ and $\overline{D C}=2 a$, hence

![](https://cdn.mathpix.com/cropped/2024_06_05_a1dcf8241dfa57377282g-09.jpg?height=257&width=393&top_left_y=244&top_left_x=1051)

$$
\frac{P_{\triangle A B C}}{P_{\triangle A C D}}=\frac{\frac{\overline{A B} \cdot h}{2}}{\frac{\overline{D C} \cdot h}{2}}=\frac{b}{a}
$$

## IX Department