B. \(\frac{{{a^3}\sqrt 3 }}{9}\).
C. \(\frac{{{a^3}}}{9}\).
D. \(\frac{{{a^3}\sqrt 2 }}{3}\).
Lời giải
Gọi \(Sx\parallel AD\parallel BC\) khi đó \(Sx = (SBC) \cap (SAD)\)
Ta có \(\left\{ \begin{array}{l}BC \bot AB\\BC \bot SA\end{array} \right. \Rightarrow BC \bot (SAB)\)
\( \Rightarrow \left\{ \begin{array}{l}AD \bot SA\\BC \bot SB\end{array} \right. \Rightarrow \left\{ \begin{array}{l}Sx \bot SA\\Sx \bot SB\end{array} \right. \Rightarrow \widehat {((SBC),(SAD))} = \widehat {ASB} = {60^^\circ }\)
\(\tan {60^^\circ } = \frac{{AB}}{{SA}} \Rightarrow AB = \frac{{SA}}{{\tan {{60}^^\circ }}} = \frac{a}{{\sqrt 3 }}\)
\({V_{S.ABCD}} = \frac{1}{3}SA \cdot {S_{ABCD}} = \frac{1}{3}\frac{{{a^2}}}{3} \cdot a = \frac{{{a^3}}}{9}\).
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