Calculate the shortest distance from one corner to another corner

A city block, 500 ft by 500 ft has a large building 300ft by 300ft in the exact center. The rest of the block is an unobstructed paved lot. What is the shortest distance from the SW corner to the NE corner of the city block, going through the paved lot?

Pivot points within a 7 point hinge

Please see the attached file for the fully formatted problem(s). My problem is explained more in my attachment, but briefly, I require some form of equation or graph to calculate where the pivot points within a seven point hinge system need to be in order for the rotating edge to rotate around a origin. The question is in regard to the hinge system. I require the hinge, when expanded, to rotate around the origin with 0 tolerance. THE HINGE What I know Through pure trial and error, involving no mathematical formulas or graphs, I have narrowed down where the pivot points, approximately need to be placed so it rotates around the origin. I did this by lengthening and shortening dist... click for more

Geometry and Algebra: Goat on a Rope Problem

How to calculate the length of the rope to which the goat is attached using geometry and algebra: A zero dimensional goat is attached by a rope to a point on the perimeter of a circular field (two dimensional). How long should the rope be (in terms of the radius of the field) so the goat can reach and graze exactly half of the grass of the field?

Prove this version of the Law of Cosines b^2 = a^2 + c^2 - 2ac cosA

Consider a triangle with sides a, b and c and angles A, B and C. Prove the following version of the Law of Cosines : b^2 = a^2 + c^2 - 2ac cosA

Inscribing a regular polygon

A clock maker wishes to make a 24 hour clock by inscribing a regular 24-gon in a circle. Determine the measure of a central angle and the measure of a vertex angle of the polygon.