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Bài 8.7 trang 63 SGK Toán 11 tập 2 - Cùng khám phá

Cho hình chóp S.ABCD có (SA bot (ABCD)) và đáy là hình vuông. Gọi M là hình chiếu vuông góc của A trên SB. Chứng minh (AM bot (SBC)) và (BD bot SC).
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Đề bài

Cho hình chóp S.ABCD có \(SA \bot (ABCD)\) và đáy là hình vuông. Gọi M là hình chiếu vuông góc của A trên SB. Chứng minh \(AM \bot (SBC)\) và \(BD \bot SC\).

Phương pháp giải - Xem chi tiết

Chứng minh \(BC \bot \left( {SAB} \right)\) từ đó suy ra \(BC \bot AM\) Chứng minh \(AM \bot \left( {SBC} \right)\) dựa vào chứng minh \(AM \bot BC\) và \(SB\) Chứng minh \(BD \bot \left( {SAC} \right)\) từ đó suy ra \(BD \bot SC\)

Lời giải chi tiết

+) Vì \(SA \bot \left( {ABCD} \right) \Rightarrow SA \bot BC\)Vì \(\left\{ \begin{array}{l}BC \bot SA\\BC \bot AB\,\,\left( {gt} \right)\end{array} \right. \Rightarrow BC \bot \left( {SAB} \right)\). Mà \(AM \subset \left( {SAB} \right) \Rightarrow BC \bot AM\)Vì \(\left\{ \begin{array}{l}AM \bot BC\\AM \bot SB\,\,\left( {gt} \right)\end{array} \right. \Rightarrow AM \bot \left( {SBC} \right)\)+) Ta có \(SA \bot \left( {ABCD} \right) \Rightarrow SA \bot BD\)Vì \(\left\{ \begin{array}{l}BD \bot SA\\BD \bot AC\,\,\left( {gt} \right)\end{array} \right. \Rightarrow BD \bot \left( {SAC} \right) \Rightarrow BD \bot SC\)

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