Here's how I would do this- probably more "sophisticated" than necessary.
Set up a coordinate system with x- axis along the center wall and origin in the middle of that center wall.
The walls of the room will be at x= -50, x= 50, y= -40, and y= 40. The four corners of the wall are at (-50, 40), (-50, -40), (50, -40) and (50, 40). The ends of the center wall are at (-35, 0) and (35, 0).
The "path" around the room makes turns at the midpoint of the line from (35, 0) TO (50, 40), the midpoint or the line from (-35, 0) to (-50, 40), the midpoint of the line from (-35, 0) to (-50, -40), and the midpoint of the line from (35, 0) to (50, -40), Those midpoints are (42.5, 20), (-42.5, 20), (-42.5, -20), and (42.5, -20). The distance from (42.5, 20) to (-42.5, 20) is 85 and the distance from (-42.5, 20) to (-42.5, -20) is 40. The other two distances are, of course, the same so the length of the circuit is 2(85+ 40)= 2(125)= 250 ft. just as you say!