Consider a triangle with side lengths 4 meters, 6 meters, and 9 meters. A dog runs around the triangle in such a way that the shortest distance of the dog from the triangle is exactly 1 meter. The total distance covered (in meters) by the dog in one round is

When the dog runs along the sides of the triangle, its path is parallel to each side. The total length of these straight sections is simply the perimeter of the triangle.

Hence, the perimeter of the triangle = 19 m

Now, at each of the three corners (vertices) of the triangle, the dog's path forms a circular arc to maintain its 1-meter distance. 

So, we can say the radius of each arc is the dog's distance from the triangle, which is 1 meter.

Therefore, the three curved arcs at the corners will make one complete circle with a radius of 1 meter.

So, the total length of this curved path is the circumference of that circle: $$2\times\pi\ \times1=2\pi\ m$$ 

Total Distance = Straight Path + Curved Path = 19 + 2π meters