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Giải bài 7.10 trang 28 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hình chóp \(S.ABCD\) có đáy \(ABCD\) là hình thoi tâm \(O\) và \(SA = SC\), \(SB = SD\).

Tổng hợp đề thi họ🃏c kì 2 lớp 11 tất cả🎉 các môn - Kết nối tri thức

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Đề bài

 Cho hình chóp \(S.ABCD\) có đáy \(ABCD\) là hình thoi tâm \(O\) và \(SA = SC\), \(SB = SD\). Chứng minh rằng a) \(SO \bot \left( {ABCD} \right)\); b) \(AC \bot \left( {SBD} \right)\) và \(BD \bot \left( {SAC} \right)\).

Phương pháp giải - Xem chi tiết

a)  Chứng minh tam giác \(SAC,SBD\) cân, \(O\) là trung điểm \(AC,BD\) từ đó suy ra \(SO \bot AC,BD \Rightarrow SO \bot \left( {ABCD} \right)\). b) Chứng minh  \(AC \bot BD,AC \bot SO\) suy ra \(AC \bot \) (SBD). Chứng minh  \(AC \bot BD,BD \bot SO\) suy ra \(AC \bot \) (SBD).

Lời giải chi tiết

a) Vì \(O\) là giao điểm của \(AC\) và \(BD\) nên \(O\) là trung điểm của \(AC\) và \(BD\)  suy ra tam giác \(SAC,SBD\) cân, suy ra \(SO \bot AC,SO \bot BD\).Do đó \(SO \bot \left( {ABCD} \right)\).b) Vì \(AC \bot BD,AC \bot SO\) nên \(AC \bot \) (SBD).Tương tự, ta được \(BD \bot \left( {SAC} \right)\).

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