Pythagorean Theorem (direct and inverse). Trapezoids (miscellaneous).

In a convex quadrilateral $M N L Q$, the angles at vertices $N$ and $L$ are right angles, and $\operatorname{tg} \angle Q M N = 2 / 3$. Find the diagonal $N Q$, if it is known that side $L Q$ is half the length of side $M N$ and 2 units longer than side $L N$.
Quadrilateral $M N L Q$ is a rectangular trapezoid with bases $L Q$ and $M N$. Let $K$ be the projection of point $Q$ on $M N$. Denote $L Q=x$. Then

$Q K=L N=x-2, K M=x, \frac{x-2}{x}=\frac{2}{3}$.

From this, $x=6$. Therefore, $N Q^{2}=N K^{2}+Q K^{2}=52$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c1015ea0b2d2805d8876g-03.jpg?height=286&width=728&top_left_y=1658&top_left_x=665)

## Answer

$2 \sqrt{13}$.