Notes
semi-circle in a square ii solution

Solution to the Semi-Circle in a Square II Puzzle

Solution by Angle in a Semi-Circle and Symmetry of a Square

With the points labelled as above, angle $A \hat{E} D$ is the angle in a semi-circle and so is $90^\circ$. Therefore, rotating the square by $90^\circ$ (either clockwise or anti-clockwise) rotates line segment $A B$ so that it is parallel to line segment $C D$. It also still runs between two sides of the square, since rotating the square leaves the square looking the same. Therefore, line segments $A B$ and $C D$ have the same length, and so the missing length is $25$.