jmc

Let us label this diagram. $△ACE$ and $△BCD$ are similar by AA since they share $\angle ACB$ and $\angle AEC$ and $\angle BDC$ are both right angles and hence congruent. So $$\frac{CE}{CD}=\frac{AC}{BC}.$$ Plugging in values, we have $$\frac{2}{3}=\frac{8}{x+2}.$$ Solving this gives $x+2=12,$ or $x=\boxed{10}.$