Arch Ratios?

There are 3 paths that Mary can take to get from A and B; The straight line segment on the bottom, a right triangle, and the outside semi-circle path. If the bottom path is 4 miles long, how long (in miles) the sum of the other two distances? If the answer can be expressed as $\sqrt{a} + b \pi$ , where $a, b$ are positive integers, find $a + b$ .

Extension: Find the ratio of the triangular and semi-circular paths. Does that ratio hold still for other lengths of the base? If the bottom path was 5 miles, would all ratios between the sides stay the same?