User:Azjps/sandbox

$[asy] defaultpen(fontsize(11pt)); size(250); real Radius = 4; pair O=(0,0), A=(Radius,0), B=A*expi(pi/3), C=((63^.5-3^.5)/2,0), D=C*expi(pi/3); draw(Circle(O,Radius)); dot(O); draw(B--O); draw(C--A); pair F=(C+(0.5*3^0.5,0.5)), G=(D+(0.5*3^0.5,0.5)); draw(C--D--G--F); draw(O--C--F--cycle,linewidth(1)); picture p = new picture; draw(p,Circle(C,0.15)); clip(p,O--C--F--cycle); add(p); p = new picture; draw(p,Circle(D,0.15)); clip(p,O--D--C--cycle); add(p); clip(currentpicture,B+(0,1)--O-(0.2,0.2)--A+(1,0)--cycle); label("$x$",(O+C)/2,S); label("$x$",(D+C)/2,NE); label("$1$",(F+C)/2,SE); label("$4$",(O+F)/2,NW); label("$150^{\circ}$",C+(-0.08,0.08),NW); label("$60^{\circ}$",D+(0,-0.15),S); [/asy]$

By the Law of Cosines on the bolded triangle, $\begin{align*} 4^2 &= x^2 + 1^2 - 2 \cdot x \cdot 1 \cos 150^{\circ}\\ 0 &= x^2 + \sqrt{3}x - 15 \\ x &= \boxed{\frac{3\sqrt{7}-\sqrt{3}}{2}} \end{align*}$

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