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Bài 4.10 trang 100 SGK Toán 11 tập 1 - Cùng khám phá

Cho tứ diện ABCD. Gọi M, N lần lượt là trung điểm của các cạnh AB, CD và G là trung điểm của đoạn MN.
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Đề bài

Cho tứ diện ABCD. Gọi M, N lần lượt là trung điểm của các cạnh AB, CD và G là trung điểm của đoạn MN. a) Tìm giao điểm A' của đường thẳng AG và mặt phẳng (BCD). b) Qua M, kẻ đường thẳng Mx song song với AA' và Mx cắt (BCD) tại M'. Chứng minh B, M', A' thẳng hằng và BM'=M'A'=A'N.

Phương pháp giải - Xem chi tiết

a) Cách tìm giao điểm của một đường thẳng a với một mặt phẳng (P): + Bước 1: Tìm \(\left( Q \right) \supset a\). Tìm \(d = \left( P \right) \cap \left( Q \right)\) + Bước 2: Tìm \(I = a \cap d\). I chính là giao điểm của a và (P). b) Chứng minh 3 điểm cùng thuộc 2 mặt phẳng phân biệt thì 3 điểm đó thẳng hàng.

Lời giải chi tiết

a)\(\left\{ \begin{array}{l}G \in MN\\MN \subset \left( {ABN} \right)\end{array} \right. \Rightarrow G \in \left( {ABN} \right)\)\( \Rightarrow AG \subset \left( {ABN} \right)\)Ta có: \(\left( {ABN} \right) \cap \left( {BCD} \right) = BN\)Trong (ABN), gọi \(AG \cap BN = A'\) \( \Rightarrow A' = AG \cap \left( {BCD} \right)\)b)\(\left\{ \begin{array}{l}Mx//AA'\\AA' \subset \left( {ABN} \right)\\M \in \left( {ABN} \right)\end{array} \right. \Rightarrow Mx \subset \left( {ABN} \right)\)Mà \(M' = Mx \cap \left( {BCD} \right)\)Suy ra \({{\rm{M}}^{\rm{'}}}\) nằm trên giao tuyến của (ABN) và (BCD) chính là đường thẳng BN.Vậy B, M’, A’ thẳng hàng.Xét tam giác \(ABA'\) có: \(\left\{ \begin{array}{l}MM'//AA'\\MA = MB\end{array} \right. \Rightarrow M'A' = M'B\)Xét tam giác \(NMM'\) có: \(\left\{ \begin{array}{l}GA//MM'\\MG = GN\end{array} \right. \Rightarrow M'A' = A'N\)\( \Rightarrow BM' = M'A' = A'N\).

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