Minimum-length roads A house is located at each corner of a square with side lengths of 1 mi. What is the length of the shortest road system with straight roads that connects all of the houses by roads (that is, a road system that allows one to drive from any house to any other house?)? (H ? int: Place two points inside the square at which roads meet.?) (S ? ource: Halmos. for Mathematicians Young and ? Old.)

Solution Step 1 Consider A BCD as the square and let the houses be located at each corner. Step 2 : Let E be the optimum point, located in th e middle of the road F G. Consider D as the sum of the individual distances to ea ch house from the point E. Therefore, As the shortest length of the roads to each house is to be found out therefore optimize the function D. In order to this, find the value of each road and after adding those find the minimum value for D . Each house is located at a distance of 1mile from its adjacent house. Therefore, Consider as the distance of IB such that . Also, Since E is located in the middle, Apply Pythagoras theorem to triangles AFH. It is clear from the diagram that, Therefore, Now substitute all the values into D,