The ratio of the measures of the angles of a triangle is 3:2:1. Given that the shortest side of the triangle is 12 meters long, what is the number of meters in the longest side of the triangle?
Since the ratio of the angle measures is $3:2:1$, the angles have measures $3x$, $2x$, and $x$ for some value of $x$.  The angle measures of a triangle sum to $180^\circ$, so we have $3x+2x+x = 180^\circ$.  Simplifying gives $6x=180^\circ$, so $x=30^\circ$.  Therefore, the angle measures are $90^\circ$, $60^\circ$, and $30^\circ$, which means the triangle is a 30-60-90 triangle.  In such a triangle, the hypotenuse has twice the length of the leg opposite the $30^\circ$ angle, so the hypotenuse of the triangle in the problem has length $2\cdot 12 = \boxed{24}$ meters.