One circle of radius 4 cm lies inside a circle of radius 10 cm, and the two circles touch at exactly one point. How far apart are their centres?

When two circles touch INTERNALLY the point of contact lies on the line of centres, on the same side of both centres, so the distance between the centres is the DIFFERENCE of the radii: 10 − 4 = 6 cm. The trap 14 cm is the sum, which is the condition for touching externally; 2 cm halves the difference; and √84 = √(100 − 16) borrows the tangent-length formula, which has nothing to do with a tangency condition.