V_{S.A′B′C′D′} = V_{S.A′B′C′} + V_{S.A′C′D′}

Lời giải

\(\frac{{SA}}{{S{A^\prime }}} + \frac{{SC}}{{S{C^\prime }}} = \frac{{SB}}{{S{B^\prime }}} + \frac{{SD}}{{S{D^\prime }}} \Rightarrow 2 + 4 = 3 + \frac{{SD}}{{S{D^\prime }}} \Rightarrow \frac{{SD}}{{S{D^\prime }}} = 3 \Rightarrow SD = 3S{D^\prime }\)

\(\frac{{{V_{S.{A^\prime }{B^\prime }{C^\prime }}}}}{{{V_{S.ABC}}}} = \frac{{S{A^\prime }}}{{SA}}.\frac{{S{B^\prime }}}{{SB}}.\frac{{S{C^\prime }}}{{SC}} = \frac{1}{2}.\frac{1}{3}.\frac{1}{4} = \frac{1}{{24}} \Rightarrow {V_{S.{A^\prime }{B^\prime }{C^\prime }}} = \frac{1}{{24}}{S_{S.ABC}}\)

\(\frac{{{V_{S.{A^\prime }{C^\prime }{D^\prime }}}}}{{{V_{S.ACD}}}} = \frac{{S{A^\prime }}}{{SA}}.\frac{{S{D^\prime }}}{{SD}}.\frac{{S{C^\prime }}}{{SC}} = \frac{1}{2}.\frac{1}{3}.\frac{1}{4} = \frac{1}{{24}} \Rightarrow {V_{S.{A^\prime }{C^\prime }{D^\prime }}} = \frac{1}{{24}}{S_{S.ACD}}\)

\( \Rightarrow {V_{S.{A^\prime }{B^\prime }{C^\prime }{D^\prime }}} = {V_{S.{A^\prime }{B^\prime }{C^\prime }}} + {V_{S.{A^\prime }{C^\prime }{D^\prime }}} = \frac{{{V_{S.ABC}} + {V_{S.ACD}}}}{{24}} \Rightarrow \frac{{{V_{S.{A^\prime }{B^\prime }{C^\prime }{D^\prime }}}}}{{{V_{S.ABCD}}}} = \frac{1}{{24}}\)