Gold Prospector Mike has found a promising vein of gold. He would like to keep the gold find for himself, but due to an error in the initial land survey, his claim diagonally overlaps with his neighbor Tom‘s claim (see sketch). A corner of Tom‘s rectangular claim is located right in the center of Mike‘s square-format claim.
In order to avoid legal disputes, Mike has agreed that Tom gets a share of the gold find. This percentage is calculated from half of the overlapping area based on the entire area of Mike's claim. What is Tom‘s share?
Rotate Tom‘s claim around the corner so that the sides of both claims are parallel to each other. Now it is clear that Tom's claim overlaps a quarter of Mike’s claim. This proportion does not vary by rotation because the lost triangle is congruent to the additional triangle. As agreed, Tom gets a share based on half of the overlapping area. Thus, Tom receives half of the quarter (1/8) and gets 12.5 % of the gold find.