Notes
semi-circle and quarter circle in a square solution

Solution to the Semi-Circle and Quarter Circle in a Square Puzzle

Solution by Symmetry and Angle Between a Radius and Tangent

In the above diagram, $G$ is the centre of the semi-circle. The line $B G$ crosses $E F$ at right-angles because $B H$ is a radius of the quarter circle and $E F$ is tangent to it, and the angle between a radius and tangent is $90^\circ$.

Therefore $E F$ and $B G$ are perpendicular line segments that both cross the square $A B C D$. By symmetry they have the same length. Since $B G$ is made up of two radii (one from each circle), it has length $8 + 5 = 13$. Thus also $E F$ has length $13$.