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Giải bài 2 trang 133 sách bài tập toán 11 - Chân trời sáng tạo tập 1

Cho hình chóp S. ABCD có đáy ABCD là hình bình hành và O là giao điểm của AC và BD. Gọi M, N, P lần lượt là ba điểm nằm trên các cạnh AB, BC, SO. Xác định giao tuyến của mặt phẳng (MNP) với các mặt của hình chóp S. ABCD (nếu có).

Tổng hợp đề thi học kì 2 lớp 11 tất cả các m𒈔ôn - Chân trời sáng tạo

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

Cho hình chóp S. ABCD có đáy ABCD là hình bình hành và O là giao điểm của AC và BD. Gọi M, N, P l💮ần lượt là ba điểm nằm trên các cạnh AB, BC, SO. Xác định giao tuyến của mặt phẳng (MNP) với các mặt của hình chóp S. ABCD (nếu có).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về giao tuyến giữa hai mặt phẳng để tìm giao tuyến: Đường thẳng d chung giữa hai mặt phẳng (P) và (Q) được gọi là giao tuyến của (P) và (Q), kí hiệu  \(d = \left( P \right) \cap \left( Q \right)\).

Lời giải chi tiết

Vì \(M \in AB,N \in BC,AB \subset \left( {ABCD} \right),BC \subset \left( {ABCD} \right) \Rightarrow MN \subset \left( {ABCD} \right)\) Lại có: \(MN \subset \left( {MNP} \right) \Rightarrow \left( {MNP} \right) \cap \left( {ABCD} \right) = MN\) Trong mặt phẳng (ABCD), gọi H là giao điểm của MN và DC, K là giao điểm của MN và AD, I là giao điểm của NO và AD. Trong mặt phẳng (SIO), gọi G là giao điểm của NP và SI. Trong (SAD), gọi T là giao điểm của KG và SA và R là giao điểm của KG và SD. Trong mặt phẳng (SCD), gọi Q là giao điểm của RH và SC. Khi đó, \(\left( {MNP} \right) \cap \left( {SAB} \right) = TM,\left( {MNP} \right) \cap \left( {SCB} \right) = NQ,\)\(\left( {MNP} \right) \cap \left( {SCD} \right) = QR,\left( {MNP} \right) \cap \left( {SAD} \right) = TR\)

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