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Bài 4.15 trang 105 SGK Toán 11 tập 1 - Cùng khám phá

Cho hình chóp S.ABCD có đáy ABCD là một tứ giác lồi. Gọi O là giao điểm của hai đường chéo AC và BD. Mặt phẳng \(\left( \alpha \right)\) đi qua O, \(\left( \alpha \right)\) song song với AB và SC.
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là một tứ giác lồi. Gọi O là giao điểm của hai đường chéo AC và BD. Mặt phẳng \(\left( \alpha  \right)\) đi qua O, \(\left( \alpha  \right)\) song song với AB và SC. Xác định giao điểm của \(\left( \alpha  \right)\) với các đường thẳng SA, AD, BC.

Phương pháp giải - Xem chi tiết

Xác định giao điểm của một mặt phẳng (P) song song với a, đi qua O và đường thẳng b: + Tìm một mặt phẳng (Q) chứa O (hoặc một điểm thuộc (P)), a, b. + Giao tuyến của (P) và (Q) là đường thẳng d đi qua O (hoặc một điểm thuộc (P)) và song song với a. + Tìm giao điểm của b và d. Đây chính là giao điểm cần tìm.

Lời giải chi tiết

Mặt phẳng \(\left( \alpha  \right)\) đi qua O và song song với AB nên \(\left( \alpha  \right)\) cắt (ABCD) theo giao tuyến d đi qua O là song song với AB. Gọi E là giao điểm của d với AD, F là giao điểm của d với BC.Vậy giao điểm của \(\left( \alpha  \right)\) với AD là E, với BC là F.Mặt phẳng \(\left( \alpha  \right)\) đi qua O và song song với SC nên \(\left( \alpha  \right)\) cắt (SAC) theo giao tuyến d đi qua O là song song với SC. Gọi G là giao điểm của d với SA.Vậy giao điểm của \(\left( \alpha  \right)\) với SA là G.

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