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Giải bài 18 trang 95 sách bài tập toán 11 - Cánh diều

Cho hình tứ diện đều ABCD. Chứng minh \(AB \bot CD.\)

🅘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 tứ diện đều ABCD. Chứng minh \(AB \bot CD.\)

Phương pháp giải - Xem chi tiết

Chứng minh hai đường thẳng vuông góc bằng cách chứng minh một đường thẳng vuông góc với một mặt phẳng chứa đường thẳng kia.

Lời giải chi tiết

Gọi I là trung điểm của CD.Vì ABCD là hình tứ diện đều nên hai tam giác ACD và BCD là các tam giác đều.Suy ra \(AI \bot CD,{\rm{ }}BI \bot CD.\)Mà AI, BI cắt nhau trong mặt phẳng (ABI) nên \(CD \bot \left( {ABI} \right).\)Mà \(AB \subset \left( {ABI} \right) \Rightarrow AB \bot CD.\)

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