Câu hỏi 28

\( + )\)\(S.ABC\) là chóp tam giác đều \( \Rightarrow \Delta ABC\) đều.

\( \Rightarrow AG = \dfrac{2}{3}.AH = \dfrac{2}{3}.\dfrac{{a\sqrt 3 }}{2} = \dfrac{{a\sqrt 3 }}{3}\)

\( + )\)Xét \(\Delta SGA\)có:

\(S{G^2} + A{G^2} = S{A^2}\)

\( \Rightarrow SG = \sqrt {2{a^2} - \dfrac{{{a^2}.3}}{9}}  = \dfrac{{a\sqrt 5 }}{{\sqrt 3 }}\)

\( \Rightarrow {R_{mcnt}} = \dfrac{{S{A^2}}}{{2.SG}} = \dfrac{{{{\left( {a\sqrt 2 } \right)}^2}}}{{2.\dfrac{{a\sqrt 5 }}{{\sqrt 3 }}}} = \dfrac{{a\sqrt {15} }}{5}\).

Chọn C