In the diagram, $AOB$ is a sector of a circle with $\angle AOB=60^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY ?$

Because $\angle AOB = 60$, $\triangle OXA$ is a 30-60-90 triangle. This means that $AX=6$ and $OX=6 \sqrt{3}$.

Then we can use power of a point on $X$ to get the equation $6\sqrt3 \cdot XY = 36$, from which we get $XY =2 \sqrt{3}$.