2021 Fermat Problem 18

In the diagram, \triangle PQR is an isosceles triangle with PQ = PR. Semi-circles with diameters PQ, QR and PR are drawn. The sum of the areas of these three semi-circles is equal to 5 times the area of the semi-circle with diameter QR. The value of \cos(\angle PQR) is

\textbf{(A)}\ \frac{1}{3}\quad \textbf{(B)}\ \frac{1}{\sqrt{8}}\quad \textbf{(C)}\ \frac{1}{\sqrt{12}}\quad \textbf{(D)}\ \frac{1}{\sqrt{15}}\quad \textbf{(E)}\ \frac{1}{\sqrt{10}}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.