In a certain triangle the circumradius measures 10 cm and the inradius measures 3 cm. How far apart are the circumcentre and the incentre?

Euler's relation for a triangle gives $O{I}^2=R(R-2r)=10\times (10-6)=40$, so $OI$ = √40 = 2√10 cm, about 6.3 cm. Notice the relation also proves $R$ ≥ $2r$ for every triangle, and 10 ≥ 6 here, so the data is consistent. The trap 7 cm subtracts the radii, 13 cm adds them, and √70 comes from writing the formula as $R(R-r)$ and forgetting that the inradius enters doubled.