In a circle of radius $10$ cm, a chord is drawn so that the perpendicular from the centre to the chord is $2$ cm shorter than the half-chord. The length of the chord is—

Let the half-chord be $h$; the perpendicular distance is $h-2$, and $(h-2)^2 + h^2 = 10^2$ gives $h^2 - 2h - 48 = 0 = (h-8)(h+6)$, so $h = 8$. The full chord is $2h = 16$ cm.