Within a farmer's pastures, there is a one-acre tract of land shaped like a right triangle. At the midpoint of the hypotenuse is a post, to which a dog is tethered with just enough rope to reach the endpoints of the hypotenuse. There are also posts at the midpoints of the legs of the triangle; to each of which is tethered a sheep with just enough rope to reach the endpoints of its respective leg of the triangle.

How much space do the sheep have to graze in (collectively) without having to worry about the dog reaching them?

The answer is 0. As the hypotenuse forms the diameter of the semicircle the dog is alotted to roam in, regardless of the dimensions of the right triangle, that right triangle will always be completely inscribed within that semicircle. Moreover, the point where the legs meet on the triangle will always lie on the circumference of the circle. A more detailed proof (Euclidean) can be obtained if required--but I hope this will be sufficient.

Posted by Benjamin J. Ladd on 2003-11-15 07:03:18