3. (15 points). On side $A B$ of parallelogram $A B C D$, a point $P$ is taken, and on side $B C$ - a point $Q$ such that $3 A B=7 B P, 3 B C=4 B Q$. Find the ratio $D O: O P$, where point $O$ is the intersection point of segments $A Q$ and $D P$.
# Solution

From the equality $3 A B=7 B P$, it follows that $A B=7 x, B P=3 x$, where $x$ is the coefficient of proportionality. Similarly, from $3 B C=4 B Q$, it follows that $B C=4 y, B Q=3 y$. Draw lines through points $P$ and $D$ parallel to $A Q$, and let points $M$ and $N$ be the points of intersection of these lines with line $B C$. Since $A Q N D$ is a parallelogram, $Q N=A D=4 y$. Apply Thales' theorem twice: $\frac{M Q}{B Q}=\frac{P A}{B A}=\frac{4}{7}$, and thus, $\frac{D O}{O P}=\frac{N Q}{Q M}=\frac{7}{3}$

![](https://cdn.mathpix.com/cropped/2024_05_06_34be784d6fd67db418cfg-2.jpg?height=320&width=1102&top_left_y=508&top_left_x=548)

Answer: $\frac{7}{3}$.

## Criteria:

| Points | Conditions for awarding |
| :---: | :--- |
| 15 points | Complete, justified solution |
| 12 points | The solution approach is correct, but a computational error has been made. |
| 10 points | Auxiliary lines have been drawn and the ratio of the lengths of the resulting segments has been found, but further actions are not performed or are performed incorrectly. |
| 5 points | One of the auxiliary lines has been drawn and the ratio of the lengths of the resulting segments has been found, but further actions are not performed or are performed incorrectly. |
| 0 points | The solution does not meet the criteria listed above |