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Giải bài 3 trang 61 sách bài tập toán 11 - Chân trời sáng tạo tập 2

Cho hình chóp S.ABC có đáy ABC là tam giác vuông cân tại B và \(SA \bot \left( {ABC} \right)\).

Tổng hợp đề thi học kì 2 lớp✨ 11 tất cả các môn - Châ🎶n trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Cho hình chóp S.ABC có đáy ABC là tam giác vuông cân tại B và \(SA \bot \left( {ABC} \right)\). a) Chứng minh rằng \(\left( {SBC} \right) \bot \left( {SAB} \right)\). b) Gọi M là trung điểm của AC. Chứng minh rằng \(\left( {SBM} \right) \bot \left( {SAC} \right)\)

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về điều kiện để hai mặt phẳng vuông góc: Điều kiện cần và đủ để hai mặt phẳng vuông góc là mặt phẳng này chứa một đường thẳng vuông góc với mặt phẳng kia.

Lời giải chi tiết

a) Vì tam giác vuông cân tại B nên \(AB \bot BC\) Lại có: \(SA \bot \left( {ABC} \right),BC \subset \left( {ABC} \right) \Rightarrow SA \bot BC\) Do đó, \(BC \bot \left( {SAB} \right)\), mà \(BC \subset \left( {SBC} \right) \Rightarrow \left( {SBC} \right) \bot \left( {SAB} \right)\) b) Vì tam giác vuông cân tại B nên BM là đường trung tuyến đồng thời là đường cao. Do đó, \(BM \bot AC\) Lại có: \(SA \bot \left( {ABC} \right),BM \subset \left( {ABC} \right) \Rightarrow SA \bot BM\) Do đó, \(BM \bot \left( {SAC} \right)\), mà \(BM \subset \left( {SBM} \right) \Rightarrow \left( {SBM} \right) \bot \left( {SAC} \right)\)

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