a,
$\widehat{SAD}=120^o$
$\Rightarrow \widehat{ABC}=180^o-120^o=60^o$
Suy ra $\Delta ABC$ đều, cạnh $a$
$S_{ABC}=\dfrac{a^2\sqrt3}{4}$
Có $\Delta ABC=\Delta ACD$ (c.c.c)
$\Rightarrow S_{ABCD}=\dfrac{a^2\sqrt3}{2}$
$SA=a$
$\Rightarrow V_{S.ABCD}=\dfrac{1}{3}a.\dfrac{a^2\sqrt3}{2}=\dfrac{a^3\sqrt3}{6}$
b,
Đặt $h=d(D, (SBC))$
$SA\bot (ABCD)$
$\Rightarrow SA\bot AB, SA\bot AC$
$\Delta SAB$ có $SB=\sqrt{SA^2+AB^2}=a\sqrt2$
$\Delta SAC$ có $SC=\sqrt{SA^2+AC^2}=a\sqrt2$
$\Rightarrow \Delta SAB$ cân tại $S$
Hạ $SH\bot BC$
$\Rightarrow BH=HC=\dfrac{a}{2}$
$SH=\sqrt{SB^2-BH^2}=\sqrt{ (a\sqrt2)^2-(\dfrac{a}{2})^2}=\dfrac{a\sqrt7}{2}$
$\Rightarrow S_{SBC}=\dfrac{1}{2}a.\dfrac{a\sqrt7}{2}=\dfrac{a^2\sqrt7}{4}$
Ta có $V=\dfrac{1}{3}.h.\dfrac{a^2\sqrt7}{2}=\dfrac{a^3\sqrt3}{6}$
$\to h=\dfrac{a\sqrt{21}}{7}$
