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

Consider a triangle with side lengths $4$ m, $6$ m, and $9$ m. The dog runs around the triangle maintaining a shortest distance of exactly $1$ m from the triangle at all times.

The dog's path consists of two types of segments:

(i) Straight parts — parallel to each side of the triangle, shifted outward by $1$ m. Each straight part has the same length as the corresponding side of the triangle.

(ii) Circular arcs — at each corner, the dog traces a circular arc of radius $1$ m to smoothly go around the vertex.

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The total length of the straight parts is simply the perimeter of the triangle:

$4 + 6 + 9 = 19$ m

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At each vertex, the dog turns through the exterior angle of the triangle. Since every triangle is a convex polygon, the sum of all exterior angles is always $360°$. This means the three circular arcs together form one complete circle of radius $1$ m.

Total arc length:

$2\pi r = 2\pi(1) = 2\pi$ m

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Total distance covered by the dog in one round:

$= 19 + 2\pi$ m

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The answer is $19 + 2\pi$ meters.