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Bài 8.16 trang 72 SGK Toán 11 tập 2 - Cùng khám phá

Cho hình chóp S.ABCD có đáy ABCD là hình vuông, hai mặt phẳng (SAB) và (SAC)
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là hình vuông, hai mặt phẳng (SAB) và (SAC) củng vuông góc với mặt đáy. Chứng minh: \(\left( {SCD} \right) \bot \left( {SAD} \right),\left( {SBC} \right) \bot \left( {SAB} \right),\left( {SBD} \right) \bot \left( {SAC} \right)\).

Phương pháp giải - Xem chi tiết

Hai mặt phẳng cắt nhau cùng vuông góc với mặt phẳng thứ ba thì giao tuyến vuông góc với mặt phẳng thứ ba. Mặt phẳng (P) chứa 2 đường thẳng cắt nhau cùng vuông góc với d thì (P) vuông góc với d. Nếu trên mặt phẳng này có chứa 1 đường thẳng vuông góc với mặt phẳng kia thì 2 mặt phẳng đó vuông góc với nhau.

Lời giải chi tiết

(SAB) và (SAC) củng vuông góc với (ABCD) nên SA vuông góc với (ABCD)Suy ra SA vuông góc với CD, BC, BDMà AD vuông góc với CDNên CD vuông góc với (SAD)Suy ra \(\left( {SCD} \right) \bot \left( {SAD} \right)\)SA vuông góc với BC và AB vuông góc với BC nên BC vuông góc với (SAB)Suy ra (SBC) vuông góc với (SAB)SA vuông góc với BD và AC vuông góc với BD nên (SAC) vuông góc với BDSuy ra (SAC) vuông góc với (SBD)

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