Notes
three overlapping squares solution

Solution to the Three Overlapping Squares Puzzle

Solution by Angle at the Centre is Twice the Angle at the Circumference

Since the diagonal of the dark blue rectangle lies along the top edge of the black square, angle $A \hat{B} C$ is $135^\circ$. The reflex angle $A \hat{O} C$ is $360^\circ - 90^\circ = 270^\circ$. Since $270^\circ$ is twice $135^\circ$, using the fact that the angle at the centre is twice the angle at the circumference, the circle centred at $O$ through $A$ and $C$ also passes through $B$. Then $O B$ has the same length as $O A$ and so has length $7$.

Solution by Invariance Principle

In this configuration, the points $A$ and $B$ coincide and so the requested length is a side of the square, which is $7$.