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a) Gọi \(O = AC \cap B{\rm{D}}\)
\(ABCD\) là hình thoi \( \Rightarrow AC \bot B{\rm{D}} \Rightarrow AO \bot B{\rm{D}}\)
\(AA' \bot \left( {ABCD} \right) \Rightarrow AA' \bot AO\)
\( \Rightarrow d\left( {B{\rm{D}},AA'} \right) = AO = \frac{1}{2}AC = \frac{{a\sqrt 3 }}{2}\)
b) Tam giác \(OAB\) vuông tại \(O\)
\(\begin{array}{l} \Rightarrow BO = \sqrt {A{B^2} - A{O^2}} = \frac{a}{2} \Rightarrow B{\rm{D}} = 2BO = a\\{S_{ABC{\rm{D}}}} = \frac{1}{2}AC.B{\rm{D}} = \frac{{{a^2}\sqrt 3 }}{2}\\{V_{ABC.A'B'C'}} = {S_{ABC{\rm{D}}}}.AA' = \frac{{3{a^3}}}{4}\end{array}\)
Đặt \(x=AA'\)
Ta có: \(\overrightarrow{AB'}=\overrightarrow{AA'}+\overrightarrow{AB}\) ; \(\overrightarrow{BD'}=\overrightarrow{BB'}+\overrightarrow{BD}=\overrightarrow{BB'}+\overrightarrow{BA}+\overrightarrow{BC}=\overrightarrow{AA'}-\overrightarrow{AB}+\overrightarrow{BC}\)
\(\Rightarrow\overrightarrow{AB'}.\overrightarrow{BD'}=\left(\overrightarrow{AA'}+\overrightarrow{AB}\right)\left(\overrightarrow{AA'}-\overrightarrow{AB}+\overrightarrow{BC}\right)\)
\(=AA'^2+\overrightarrow{AA'}\left(-\overrightarrow{AB}+\overrightarrow{BC}\right)+\overrightarrow{AB}.\overrightarrow{AA'}-AB^2+\overrightarrow{AB}.\overrightarrow{BC}\)
\(=x^2-a^2+AB.BC.cos120^0\)
\(=x^2-a^2-\dfrac{a^2}{2}=x^2-\dfrac{3a^2}{2}=0\)
\(\Rightarrow x=\dfrac{a\sqrt{6}}{2}\)
\(V=\dfrac{a\sqrt{6}}{2}.2.\dfrac{a^2\sqrt{3}}{4}=\dfrac{3a^3\sqrt{2}}{4}\)
Chọn đáp án B
Gọi O = AC ∩ BD.Từ giả thiết suy ra A'O ⊥ ABCD
Cũng từ giả thiết, suy ra ABC là tam giác đều nên
Đường cao khối hộp
\(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\Rightarrow\overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB}=\overrightarrow{b}-\overrightarrow{a}\)
Theo Talet: \(\dfrac{A'K}{IK}=\dfrac{B'I}{A'D'}=\dfrac{1}{2}\Rightarrow A'K=\dfrac{2}{3}A'I\)
\(\Rightarrow\overrightarrow{A'K}=\dfrac{2}{3}\overrightarrow{A'I}=\dfrac{2}{3}\left(\overrightarrow{A'B'}+\overrightarrow{B'I}\right)=\dfrac{2}{3}\left(\overrightarrow{A'B'}+\dfrac{1}{2}\overrightarrow{B'C'}\right)\)
\(=\dfrac{2}{3}\overrightarrow{AB}+\dfrac{1}{3}\overrightarrow{BC}=\dfrac{2}{3}\overrightarrow{a}+\dfrac{1}{3}\left(\overrightarrow{b}-\overrightarrow{a}\right)=\dfrac{1}{3}\overrightarrow{a}+\dfrac{1}{3}\overrightarrow{b}\)
\(\Rightarrow\overrightarrow{DK}=\overrightarrow{DD'}+\overrightarrow{D'A'}+\overrightarrow{A'K}=\overrightarrow{AA'}-\overrightarrow{BC}+\overrightarrow{A'K}\)
\(=\overrightarrow{c}-\left(\overrightarrow{b}-\overrightarrow{a}\right)+\dfrac{1}{3}\overrightarrow{a}+\dfrac{1}{3}\overrightarrow{b}\)
\(=\dfrac{4}{3}\overrightarrow{a}-\dfrac{2}{3}\overrightarrow{b}+\overrightarrow{c}\)
