jmc

By the Pythagorean theorem, we have $$\begin{array}{rl}{9}^2+x^2& =(x+1{)}^2⟹\\ 81+x^2& =x^2+2x+1⟹\\ 2x& =80⟹\\ x& =40,\end{array}$$where $x$ is the shorter missing side. It follows that the sides of the triangle are 9, 40, and 41 meters, and the perimeter of the triangle is $9+40+41=\boxed{90}$ meters.

Note: for any odd integer $n$, the two integers closest to $n^2/2$ along with $n$ form a Pythagorean triple.