$S\overset{\vartriangle }{\mathop{ABC}}\,=\frac{1}{2}BC.AH\Rightarrow 12=\frac{1}{2}(BC)(4)$ $\Rightarrow 12=2BC\Rightarrow BC=6cm$ $\Rightarrow BH=HC=3cm$ $\overset{\vartriangle }{\mathop{AHC}}\,\to A{{C}^{2}}=A{{H}^{2}}+H{{C}^{2}}$ $\Rightarrow A{{C}^{2}}={{4}^{2}}+{{(3)}^{2}}\Rightarrow A{{C}^{2}}=25\Rightarrow AC=5cm$