9.3. In triangle $M P Q$, a line parallel to side $M Q$ intersects side $M P$, median $M M_{1}$, and side $P Q$ at points $D, E$, and $F$ respectively. It is known that $D E=5$, and $E F=7$. What is the length of $M Q$?
Answer: 17.

Solution: Draw a line through point $E$ parallel to $P Q$ ( $K$ and $L$ - the points of intersection of this line with sides $M P$ and $M Q$ respectively). Since $M M_{1}$ is a median, then $L E = E K$, and since $D F \parallel M Q$, it follows that $D E$ is the midline of triangle $M K L$. Therefore, $M L = 2 D E = 10$. Moreover, $E L \parallel F Q$ and $E F \parallel L Q$, so $L E F Q$ is a parallelogram, from which $L Q = E F = 7$. Thus, $M Q = M L + L Q = 10 + 7 = 17$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d04205b7a007d184e39cg-2.jpg?height=534&width=779&top_left_y=533&top_left_x=1141)

## Criteria:

7 points - correct answer and complete correct solution;

$3-5$ points - partially correct solution;

answer only - 0 points.