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Bài 8.1 trang 54 SGK Toán 11 tập 2 - Cùng khám phá

Cho tứ diện đều \(ABCD\). Gọi \(M\) là trung điểm của \(BC\). Tính \(\cos \left( {AB,DM} \right)\)
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Đề bài

Cho tứ diện đều \(ABCD\). Gọi \(M\) là trung điểm của \(BC\). Tính \(\cos \left( {AB,DM} \right)\)

Phương pháp giải - Xem chi tiết

+ Gọi \(N\) là trung điểm của \(AC\). Khi đó \(MN//AB\) + Góc giữa \(\left( {AB,MD} \right) = \left( {MN,MD} \right)\) + Tính các cạnh \(MN,ND,MD\) + Tính \(\cos M = \frac{{M{N^2} + M{D^2} - N{D^2}}}{{2MN.MD}}\)

Lời giải chi tiết

Giả sử tứ diện đều có cạnh bằng \(a\)Gọi \(N\) là trung điểm của \(AC\). Suy ra \(MN\) là đường trung bình của tam giác\( \Rightarrow MN//AB;MN = \frac{1}{2}AB = \frac{a}{2}\)Vì \(MN//AB\)\( \Rightarrow \left( {AB,MD} \right) = \left( {MN,MD} \right) = \widehat {NMD}\) (vì góc \(\widehat {NMD}\) là góc nhọn)Vì tam giác \(BCD\) đều nên \(MD \bot BC\)\( \Rightarrow MD = \sqrt {B{D^2} - B{M^2}} \)\( = \sqrt {{a^2} - {{\left( {\frac{a}{2}} \right)}^2}}  = \frac{{a\sqrt 3 }}{2}\)Tương tự, \(ND = \frac{{a\sqrt 3 }}{2}\)Xét \(\Delta MND\) có \(\cos M = \frac{{M{N^2} + M{D^2} - N{D^2}}}{{2MN.MD}}\)\( = \frac{{{{\left( {\frac{a}{2}} \right)}^2} + {{\left( {\frac{{a\sqrt 3 }}{2}} \right)}^2} - {{\left( {\frac{{a\sqrt 3 }}{2}} \right)}^2}}}{{2.\frac{a}{2}.\frac{{a\sqrt 3 }}{2}}} = \frac{{\sqrt 3 }}{6}\)\( \Rightarrow \widehat M \approx {73^o}\). Vậy \(\left( {AB,MD} \right) \approx {73^o}\)

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