a, \(V_{SACD}=\dfrac{1}{3}S_{ACD}\cdot SA\)
\(S_{ACD}=\dfrac{1}{2}a^2\cdot sin90^o=\dfrac{a^2}{2}\)
\(\Rightarrow V_{SACD}=\dfrac{1}{3}\cdot\dfrac{a^2}{2}\cdot a\sqrt{3}=\dfrac{a^3\sqrt{3}}{6}\)
b, Từ O dựng OM // SB
\(\Rightarrow\left(\widehat{SB,AC}\right)=\left(\widehat{OM,OC}\right)\)
Gọi \(\widehat{COM}=\alpha\)
Xét \(\Delta\) \(OMC\) : \(OC=\dfrac{1}{2}AC=\dfrac{a\sqrt{2}}{2}\)
\(OM=\dfrac{1}{2}SB\)
Xét \(\Delta\) \(SAB\) có : \(SB^2=SA^2+AB^2=3a^2+a^2=4a^2\)
\(\rightarrow SB=2a\rightarrow OM=a\)
CM là đường trung tuyến của \(\Delta\) \(SCD\) :
\(CM^2=\dfrac{SC^2+CD^2}{2}=\dfrac{SD^2}{4}\)
\(SC^2=5a^2\) ; \(SD^2=4a^2\)
\(\Rightarrow CM=\dfrac{5a^2+a^2}{2}-\dfrac{4a^2}{4}=2a^2\)
\(\Rightarrow CM=a\sqrt{2}\)
Xét \(\Delta\) OMC có :
\(CM^2=OM^2+OC^2-2OM\cdot OC\cdot cos\alpha\)
\(\Leftrightarrow2a^2=a^2+\dfrac{a^2}{2}-2a\cdot\dfrac{a\sqrt{2}}{2}\cdot cos\alpha\)
\(\Rightarrow cos\alpha=\dfrac{-1}{2\sqrt{2}}< 0\)
\(\Rightarrow cos\left(\widehat{OC,OM}\right)=\dfrac{1}{2\sqrt{2}}=cos\left(\widehat{SB,AC}\right)\)
