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Bài 4.37 trang 125 SGK Toán 11 tập 1 - Cùng khám phá

Cho hình hộp ABCD.A'B'C'D'. Gọi \(O = AC \cap BD\) và \(O' = A'C' \cap B'D'\). Gọi M, N lần lượt là trung điểm của AB và CD.
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Đề bài

Cho hình hộp ABCD.A'B'C'D'. Gọi \(O = AC \cap BD\) và \(O' = A'C' \cap B'D'\). Gọi M, N lần lượt là trung điểm của AB và CD. Ảnh của tam giác C'MN qua phép chiếu song song trên mặt phẳng (ABCD) theo phương AO' là A. Đoạn thẳng MN. B. Tam giác OBC. C. Tam giác CMN. D. Đoạn thẳng BD.

Phương pháp giải - Xem chi tiết

A' là hình chiếu song song của điểm A trên (P) theo phương d thì AA' // d và AA' cắt (P) tại A'.

Lời giải chi tiết

M, N thuộc (ABCD) nên M, N là hình chiếu song song của chính nó trên (ABCD) theo phương AO'Ta có: AA' // CC' (cùng // BB') và AA' = CC' (cùng = CC') nên ACC'A' là hình bình hành. Suy ra AC // A'C' hay AO // O'C' (1)O, O' lần lượt là giao điểm các đường chéo của hình bình ABCD, A'B'C'D' nên O là trung điểm AC, A'C'. Suy ra AO = O'C' (2)Từ (1), (2) suy ra AOC'O' là hình bình hành. Suy ra AO' // OC'Vậy O là hình chiếu song song của C' trên (ABCD) theo phương AO'Suy ra hình chiếu song song của tam giác C'MN trên (ABCD) theo phương AO' là đoạn thẳng MN.Chọn đáo án A.

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