1. **Problem statement:** We are given a triangle \(\triangle PQR\) with point \(O\) as the intersection of the perpendicular bisectors of its sides. It is given that \(OP = OQ = OR = 10\). We need to find the radius of the circumcircle of \(\triangle PQR\). 2. **Key concept:** The point where the perpendicular bisectors of a triangle meet is called the circumcenter. The circumcenter is equidistant from all vertices of the triangle. 3. **Formula:** The radius \(R\) of the circumcircle is the distance from the circumcenter \(O\) to any vertex of the triangle. Mathematically, \(R = OP = OQ = OR\). 4. **Given:** \(OP = OQ = OR = 10\). 5. **Conclusion:** Since the circumcenter is equidistant from all vertices, the radius of the circumcircle is \(10\). **Final answer:** The radius of the circumcircle of \(\triangle PQR\) is \(\boxed{10}\).