Arch Ratios?

Geometry Level 2

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?

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