Gọi \(H\) là trung điểm của \(AB\). Vì tam giác \(SAB\) cân tại \(S\) nên \(SH \bot AB\)

Ta có: \(\left\{ {\begin{array}{*{20}{l}}{\left( {SAB} \right) \bot \left( {ABCD} \right) = AB}\\{SH \subset \left( {SAB} \right),{\mkern 1mu} {\mkern 1mu} SH \bot AB}\end{array}} \right. \Rightarrow SH \bot \left( {ABCD} \right)\)

Gọi \(K\) là trung điểm của \(CD\) ta có

\(\left\{ {\begin{array}{*{20}{l}}{CD \bot HK}\\{CD \bot SH}\end{array}} \right. \Rightarrow CD \bot \left( {SHK} \right) \Rightarrow CD \bot SK\)

\(\left\{ {\begin{array}{*{20}{l}}{\left( {SCD} \right) \cap \left( {ABCD} \right) = CD}\\{SK \subset \left( {SCD} \right),{\mkern 1mu} {\mkern 1mu} \,SK \bot CD}\\{HK \subset \left( {ABCD} \right),{\mkern 1mu} {\mkern 1mu} \,HK \bot CD}\end{array}} \right.\)\( \Rightarrow \widehat {\left( {\left( {SCD} \right),\left( {ABCD} \right)} \right)} = \widehat {\left( {SK,HK} \right)} = \widehat {SKH} = \varphi \)

Vì \(AH\,{\rm{//}}\,CD \Rightarrow AH\,{\rm{//}}\,\left( {SCD} \right) \Rightarrow d\left( {A,\left( {SCD} \right)} \right) = d\left( {H,\left( {SCD} \right)} \right)\)

Trong \(\left( {SHK} \right)\) kẻ \(HI \bot SK{\mkern 1mu} {\mkern 1mu} \left( {I \in SK} \right)\) ta có: \(\left\{ {\begin{array}{*{20}{l}}{HI \bot SK}\\{HI \bot CD{\mkern 1mu} {\mkern 1mu} \left( {CD \bot \left( {SHK} \right)} \right)}\end{array}} \right. \Rightarrow HI \bot \left( {SCD} \right)\)

\( \Rightarrow d\left( {H,\left( {SCD} \right)} \right) = HI\).

Xét tam giác vuông \(HIK\) ta có \(\sin \varphi = \sin \widehat {SKH} = \frac{{HI}}{{HK}}\) \( \Rightarrow HI = HK.\sin \varphi = 2a.\frac{{\sqrt 5 }}{5} = \frac{{2a\sqrt 5 }}{5}\)

Vậy \(d\left( {A,\left( {SCD} \right)} \right) = \frac{{2a\sqrt 5 }}{5}\). Đáp án: \(\frac{{2\sqrt 5 a}}{5}\).