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Bài 8.9 trang 63 SGK Toán 11 tập 2 - Cùng khám phá

Cho hình chóp S.ABCD có đáy ABCD là hình thoi, O là giao điểm của AC và BD và SA=SC, SB= SD.
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là hình thoi, O là giao điểm của AC và BD và SA=SC, SB= SD. Chứng minh \(BC \bot SO\) và \(SC \bot BD\).

Phương pháp giải - Xem chi tiết

Chứng minh \(SO \bot AC,SO \bot BD\) từ đó suy ra \(SO \bot \left( {ABCD} \right)\) và suy ra \(SO \bot BC\) Chứng minh \(BD \bot \left( {SAC} \right)\) từ đó suy ra \(BD \bot SC\)

Lời giải chi tiết

+) Vì \(ABCD\) là hình thoi nên \(AC \bot BD\) tại trung điểm \(O\) của mỗi đườngVì \(SA = SC \Rightarrow \Delta SAC\) cân tại \(S\) và \(O\) là trung điểm \(AC\) nên \(SO \bot AC\)Vì \(SB = SD\)\( \Rightarrow \Delta SBD\) cân tại \(S\) và \(O\) là trung điểm \(BD\) nên \(SO \bot BD\)Ta có \(\left\{ \begin{array}{l}SO \bot AC\\SO \bot BD\end{array} \right. \Rightarrow SO \bot \left( {ABCD} \right) \Rightarrow SO \bot BC\)+) Vì \(SO \bot \left( {ABCD} \right) \Rightarrow SO \bot BD\)Ta có \(\left\{ \begin{array}{l}SO \bot BD\\AC \bot BD\end{array} \right. \Rightarrow BD \bot \left( {SAC} \right) \Rightarrow BD \bot SC\)

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