Đáp án:
$\begin{array}{l}
a)Xét:\Delta ADI;\Delta AHI:\\
+ AI\,chung\\
+ \widehat {AID} = \widehat {AIH} = {90^0}\\
+ HI = DI\\
\Rightarrow \Delta ADI = \Delta AHI\left( {c - g - c} \right)\\
b)Do:\Delta ADI = \Delta AHI\\
\Rightarrow AD = AH;\widehat {IAD} = \widehat {IAH}\\
Xét:\Delta ABD:\Delta ABH:\\
+ AD = AH\\
+ \widehat {BAD} = \widehat {BAH}\\
+ AB\,chung\\
\Rightarrow \Delta ABD = \Delta ABH\left( {c - g - c} \right)\\
\Rightarrow \widehat {ADB} = \widehat {AHB} = {90^0}\\
\Rightarrow AD \bot BD\\
c)\Delta AHB \sim \Delta CHA\left( {g - g} \right)\\
\Rightarrow \dfrac{{AH}}{{HC}} = \dfrac{{HB}}{{AH}}\\
\Rightarrow A{H^2} = HB.HC = 9.16 = 144\\
\Rightarrow AH = 12\left( {cm} \right)
\end{array}$