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Bài 4.18 trang 114 SGK Toán 11 tập 1 - Cùng khám phá

Cho hình chóp S.ABCD có đáy ABCD là hình bình hành tâm O. Gọi M, N lần lượt là trung điểm của SA và SD.
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là hình bình hành tâm O. Gọi M, N lần lượt là trung điểm của SA và SD. a) Chứng minh rằng (OMN) // (SBC).          b) Gọi P, Q lần lượt là trung điểm của AB, ON. Chứng minh rằng PQ // (SBC).

Phương pháp giải - Xem chi tiết

a) Nếu mặt phẳng (P) chứa 2 đường thẳng cắt nhau a, b và a, b cùng song song với mặt phẳng (Q) thì (P) song song với (Q). b) Nếu (P) và (Q) song song với nhau thì mọi đường thẳng trong (P) song song với (Q) và ngược lại.

Lời giải chi tiết

a) O là tâm hình bình hành ABCD. Suy ra O là trung điểm của AC và BDXét tam giác SAD có M, N lần lượt là trung điểm của SA, SD nên MN // ADMà AD // BC (Do ABCD là hình bình hành) nên MN // BCSuy ra MN // (SBC) (1)Xét tam giác SAC có M, O lần lượt là trung điểm của SA, AC nên MO // SCSuy ra MO // (SBC) (2)Từ (1) và (2) suy ra (MNO) // (SBC)b) Xét tam giác ABC có O, P lần lượt là trung điểm của AC, AB nên OP // BCSuy ra OP // (SBC) (3)Xét tam giác SBD có O, N lần lượt là trung điểm của BD, SD nên ON // SBSuy ra ON // (SBC) hay OQ // (SBC) (4)Từ (3) và (4) suy ra (OPQ) // (SBC)Nên PQ // (SBC).

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