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Since triangle $ABC$ is equilateral, $PX$ is also a median, bisecting side $QR$. Thus, $XR=\frac{9}{2}.$ Note that triangle $YXR$ is a 30-60-90 triangle with right angle at $Y$ and 30 degree angle at $\angle{YXR}$. Thus, $YR = \frac{\frac{9}{2}}{2} = \frac{9}{4}.$ Since the side opposite the $60^\circ$ angle in a 30-60-90 triangle is $\sqrt{3}$ times the side opposite the $30^\circ$ angle, we have $XY = \frac{9}{4} \cdot \sqrt{3} = \boxed{\frac{9\sqrt{3}}{4}}.$