Question

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

• A. \( 22 - 2\pi \)
• B. \( 22 \)
• C. \( 19 + 2\pi \)
• D. \( 22 + 2\pi \)

Hint:

This principle applies to any convex polygon, not just a triangle. The distance covered by an object moving at a constant distance 'r' around a convex polygon with perimeter 'P' is always \( P + 2\pi r \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The path of the dog consists of straight sections parallel to the sides of the triangle and curved sections around the vertices. We need to calculate the length of both parts and add them together.
Step 2: Key Formula or Approach:
1. The total distance is the sum of the lengths of the straight paths and the curved paths. 2. The straight paths are parallel to the sides of the triangle and have the same lengths. Their total length is the perimeter of the triangle. 3. The curved paths at the vertices are circular arcs. The sum of the exterior angles of any convex polygon is 360\(^\circ\) or \(2\pi\) radians. These arcs will combine to form a full circle.
Step 3: Detailed Explanation:
The dog's path can be visualized in two parts:

Straight Segments: As the dog moves along the sides of the triangle, it stays 1 meter away. This creates three straight paths parallel to the sides of the triangle. The lengths of these paths are equal to the lengths of the sides of the triangle. Total length of straight segments = Perimeter of the triangle. \[ \text{Perimeter} = 4 + 6 + 9 = 19 \text{ meters} \]
Curved Segments: As the dog moves around each vertex of the triangle, it traces a circular arc of radius 1 meter to maintain the constant distance. The angle of each arc corresponds to the exterior angle at that vertex. The sum of the exterior angles of any triangle is always 360\(^\circ\) or \(2\pi\) radians. Therefore, the three curved paths at the vertices, when put together, form a complete circle with a radius of 1 meter. The length of this combined curved path is the circumference of this circle. \[ \text{Circumference} = 2 \pi r = 2 \pi (1) = 2\pi \text{ meters} \]
The total distance covered by the dog is the sum of the lengths of the straight and curved segments. \[ \text{Total Distance} = \text{Perimeter} + \text{Circumference} = 19 + 2\pi \] Step 4: Final Answer:
The total distance covered by the dog is \( 19 + 2\pi \) meters.