$3+$ [ Trapezoids (miscellaneous)

In trapezoid $A B C D$, sides $A B$ and $C D$ are parallel and $C D=2 A B$. Points $P$ and $Q$ are chosen on sides $A D$ and $B C$ respectively such that $D P: P A=2, B Q: Q C=3: 4$. Find the ratio of the areas of quadrilaterals $A B Q P$ and $C D P Q$.

#
Extend the lateral sides of the trapezoid to their intersection.

## Solution

Let the extensions of the lateral sides $A D$ and $B C$ of the trapezoid intersect at point $K$. From the problem statement, it follows that $A B$ is the midline of triangle $K D C$. Denote $A B=z, B Q=3 t, S_{\triangle \text { AKB }}=s$. Then

$$
\begin{gathered}
D P=2 z, A K=A D=3 z, C Q=4 t, B K=B C=7 z, \\
S_{\Delta \mathrm{KDC}}=4 s, S_{\Delta \mathrm{KPQ}}=\frac{K P}{K D} \cdot \frac{K Q}{K C} \cdot S_{\Delta \mathrm{KDC}}=\frac{4}{6} \cdot \frac{10}{14} \cdot 4 s=\frac{40}{21} s, \\
S_{\mathrm{ABQP}}=S_{\Delta \mathrm{KPQ}}-S_{\Delta \mathrm{AKB}}=\frac{40}{21} s-s=\frac{19}{21} s, \\
S_{\mathrm{CDPQ}}=S_{\mathrm{ABCD}}-S_{\mathrm{ABQP}}=3 s-\frac{19}{21}=\frac{44}{21} s .
\end{gathered}
$$

Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_aa2960437c2efca8c53fg-22.jpg?height=660&width=504&top_left_y=1402&top_left_x=777)

## Answer

$\frac{19}{44}$.