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Giải bài 4 trang 94 sách bài tập toán 11 - Cánh diều

Cho tứ diện \(ABCD\). Gọi \(M,{\rm{ }}N\) lần lượt là trung điểm của \(AB,{\rm{ }}CD\).

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

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Đề bài

Cho tứ diện \(ABCD\). Gọi \(M,{\rm{ }}N\) lần lượt là trung điểm của \(AB,{\rm{ }}CD\). Chứng minh rằng bốn điểm \(M,{\rm{ }}N,{\rm{ }}C,{\rm{ }}D\) không cùng nằm trong một mặt phẳng.

Phương pháp giải - Xem chi tiết

Giả sử 4 điểm \(M\), \(N\), \(C\), \(D\) cùng nằm trong một mặt phẳng. Từ đó chứng minh rằng \(M \in \left( {BCD} \right)\), suy ra \(A \in \left( {BCD} \right)\) và suy ra điều vô lí.

Lời giải chi tiết

Do \(N\) là trung điểm của \(BC\), nên 4 điểm \(B\), \(N\), \(C\), \(D\) cùng nằm trong mặt phẳng.Giả sử 4 điểm \(M\), \(N\), \(C\), \(D\) cùng nằm trong một mặt phẳng.Điều này có nghĩa là \(M \in \left( {NCD} \right)\).Do bốn điểm \(B\), \(N\), \(C\), \(D\) cùng nằm trong mặt phẳng, ta suy ra \(M \in \left( {BCD} \right)\).Điểm \(M\) và điểm \(B\) cùng nằm trong mặt phẳng \(\left( {BCD} \right)\), nên \(BM \subset \left( {BCD} \right)\).Mặt khác, do \(M\) là trung điểm của \(AB\), nên \(A \in BM\).Suy ra \(A \in \left( {BCD} \right)\). Điều này là vô lí do \(ABCD\) là tứ diện nên bốn điểm \(A\), \(B\), \(C\), \(D\) không cùng nằm trong một mặt phẳng.

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