CPM Homework Help : APCALC Problem 10-89

The picture at the right shows a housing complex consisting of four houses. Each house sits at the corner of a square with sides that measure $100$ meters. The interior lines show roads connecting the houses. For what value of $x$ , $0 ≤ x ≤ 100$ , will the total length of the interior roads be the smallest?

Hint:

Added to the rectangle, vertical segment from left endpoint of horizontal segment, perpendicular to top side, labeled 50, creating right triangle with horizontal leg labeled 100 minus x all divided by 2.

More Help:

Total road length:

$50+4\sqrt{50^2+\Big(\frac{100-x}{2}\Big)^2}$

Rectangle, house at each vertex, 5 segments, 4 from each vertex to the fifth horizontal segment, in the center, labeled x, so that top & bottom left diagonal segments meet at the left endpoint of horizontal segment, & top & bottom right diagonal segments, meet at the right endpoint of horizontal segment