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Giải bài 16 trang 95 sách bài tập toán 11 - Cánh diều

Cho hình chóp S.ABC thoả mãn SA = SB = SC.

🌸Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

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

Cho hình chóp S.ABC thoả mãn SA = SB = SC. Gọi O là tâm đường tròn ngoại tiếp của tam giác ABC. Chứng minh rằng \(SO \bot \left( {ABC} \right).\)

Phương pháp giải - Xem chi tiết

Có duy nhất một đường thẳng đi qua một điểm cho trước và vuông góc với một mặt phẳng cho trước.

Lời giải chi tiết

Gọi \(O'\) là hình chiếu của S trên (ABC). Khi đó, \(SO' \bot \left( {ABC} \right).\)Mà \(O'A,{\rm{ }}O'B,{\rm{ }}O'C\) đều nằm trên (ABC) nên \(SO' \bot O'A,{\rm{ }}SO' \bot O'B,{\rm{ }}SO' \bot O'C.\)Xét ba tam giác \(SO'A,{\rm{ }}SO'B,{\rm{ }}SO'C\) vuông tại \(O'\) có SA = SB = SC và \(SO'\) chung nên ba tam giác đó bằng nhau. Do đó, \(O'A = O'B = O'C.\)Suy ra \(O'\) là tâm đường tròn ngoại tiếp của tam giác ABC hay \(O'\) trùng O.Vậy \(SO \bot \left( {ABC} \right).\)

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