Chọn C
Ta có: \(AK=\sqrt{\frac{AI^{2} +AB^{2} }{2} -\frac{IB^{2} }{4} } =\frac{a\sqrt{7} }{4} \Rightarrow SK=\frac{a\sqrt{21} }{4} .\)
Đặt hệ trục tọa độ Ixyznhư hình vẽ:\( I\left(0;0;0\right);A\left(-\frac{a}{2} ;0;0\right); S\left(0;\frac{a\sqrt{3} }{4} ;\frac{a\sqrt{21} }{4} \right);C\left(\frac{a}{2} ;0;0\right);\)
\(\[M\left(-\frac{a}{10} ;\frac{2a\sqrt{3} }{5} ;0\right);{\rm \; }B\left(0;\frac{a\sqrt{3} }{2} ;0\right).\] \)
Ta có:
\(\overrightarrow{AS}=\left(\frac{a}{2} ;\frac{a\sqrt{3} }{4} ;\frac{a\sqrt{21} }{4} \right) cùng phương \overrightarrow{u}=\left(2;\sqrt{3} ;\sqrt{21} \right).\)
\(\overrightarrow{MC}=\left(\frac{3a}{5} ;\frac{-2a\sqrt{3} }{5} ;0\right) cùng phương \overrightarrow{v}=\left(\sqrt{3} ;-2;0\right)\)
\(\overrightarrow{n}=\left[\overrightarrow{u},\overrightarrow{v}\right]=\left(2\sqrt{21} ;3\sqrt{7} ;-7\right) cùng phương \overrightarrow{m}=\left(2\sqrt{3} ;3;-\sqrt{7} \right); \overrightarrow{AC}=\left(a;0;0\right)\)
Vậy \(d\left(SA,CM\right)=\frac{\left|\overrightarrow{m}.\overrightarrow{AC}\right|}{\left|\overrightarrow{m}\right|} =\frac{a\sqrt{21} }{7} .\)