The Dog's Span I

Geometry Level 4

A dog is tied on a triangular pillar with each side measuring 5 meters. The pillar is attached to the corner of a perpendicular fence such that the side of the pillar farthest from where the dog is tied is perpendicular to the fence nearby. If the chain is 12 m long, what is the total span of area (in square meters) that the dog can reach? Assume the dog's size to be infinitesimal, and that the chain is not elastic. Also assume that the dog cannot pass through the point attaching the pillar to the fence.

The total area can be expressed in the mixed fraction form \(\large {a \frac {b}{c}\pi }\), where \(a\), \(b\), and \(c\) are all positive integers with \(b\) and \(c\) coprime. Express your answer in the form \(a + b + c\).

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