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Bài 4 trang 104 SGK Toán 11 tập 1 - Cánh Diều

Cho hình chóp S.ABCD có đáy ABCD là hình bình hành. Gọi M, N lần lượt là trung điểm của AB và CD. Chứng minh rằng đường thẳng MN song song với giao tuyến d của hai mặt phẳng (SBC) và (SAD).

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

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là hình bình hành. Gọi M, N lần lượt là trung điểm của AB và CD. Chứng minh rằng đường thẳng MN song song với giao tuyến d của hai mặt phẳng (SBC) và (SAD).

Phương pháp giải - Xem chi tiết

Cho đường thẳng a song song với mặt phẳng (P). Nếu mặt phẳng (Q) chứa a và cắt (P) theo giao tuyến b thì b song song với a

Lời giải chi tiết

🅠Ta có: S ∈ (SAD) và S ∈ (SBC) nên S là giao điểm của (SAD) và (SBC).

Lại có: AD // BC (do ABCD là hình bình hành);

            AD ⊂ (SAD);

            BC ⊂ (SBC).

𒐪Do đó giao tuyến d của hai mặt phẳng (SAD) và (SBC) là đường thẳng đi qua S và song song với AD, BC.

🐠Vì M, N lần lượt là trung điểm của AB và CD nên MN là đường trung bình

Do đó MN // BC // AD.

Ta có: MN // BC mà BC ⊂ (SBC) nên MN // (SBC);

𒀰           MN // AD mà AD ⊂ (SAD) nên MN // (SAD).

Có: MN // (SBC);

       MN // (SAD);

       (SAD) ∩ (SBC) = d

Suy ra MN // d.

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Tham Gia Group Dành Cho Lớp 11 Chia Sẻ, Trao Đổi Tài Liệu Miễn Phí

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