3470.51 – A Triangle in a Circle

If $AB$ and $CD$ are perpendicular diameters of circle $Q$ , and angle $\angle QPC = 60^\circ$ , then what is the ratio of $PQ$ to $AQ$ ?

Solution

$\dfrac{PQ}{CQ}=\dfrac{1}{\sqrt{3}}$ , since we are looking at a 30-60 right triangle. Since $AQ = CQ$ , $\dfrac{PQ}{AQ}=\dfrac{1}{\sqrt3}=\dfrac{\sqrt3}{3}$ .