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Bài 4.17 trang 114 SGK Toán 11 tập 1 - Cùng khám phá

Cho hình chóp S.ABCD có đáy là hình thang ABCD với đáy lớn là AD, AD = 2BC. Gọi I, K, L lần lượt là trung điểm của đoạn AD, SA, SD. Chứng minh rằng (SAB) // (ILC) và (SCD) // (BIK).
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Đề bài

Cho hình chóp S.ABCD có đáy là hình thang ABCD với đáy lớn là AD, AD = 2BC. Gọi I, K, L lần lượt là trung điểm của đoạn AD, SA, SD. Chứng minh rằng (SAB) // (ILC) và (SCD) // (BIK).

Phương pháp giải - Xem chi tiết

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).

Lời giải chi tiết

Xét tam giác SAD có I, L lần lượt là trung điểm của AD, SD nên IL // SA. Suy ra IL // (SAB) (1)AD = 2BC mà I là trung điểm AD nên AI = BC, AI // BC (do ABCD là hình thang). Suy ra AICB là hình bình hành\( \Rightarrow \)IC // AB \( \Rightarrow \)IC // (SAB) (2)Từ (1) và (2) suy ra  (ILC) // (SAB)Xét tam giác SAD có I, K lần lượt là trung điểm của AD, SA nên IK // SD. Suy ra IK // (SCD) (3)AD = 2BC mà I là trung điểm AD nên ID = BC, ID // BC (do ABCD là hình thang). Suy ra BIDC là hình bình hành\( \Rightarrow \)BI // CD \( \Rightarrow \)BI // (SCD) (4)Từ (3) và (4) nên (BIK) // (SCD).

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