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Giải bài 4.61 trang 74 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hình lăng trụ tứ giác ABCD.A’B’C’D’

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Đề bài

Cho hình lăng trụ tứ giác ABCD.A’B’C’D’. Gọi M, N, M’, N’ lần lượt là trung điểm của các cạnh AB, CD, A’B’, C’D’. a) Chứng minh rằng bốn điểm M, N, M’, N’ đồng phẳng và tứ giác MNN’M’ là hình bình hành b) Giả sử MN không song song với BC. Xác định giao tuyến của hai mặt phẳng (MNN’M’) và (BCC’B’).

Phương pháp giải - Xem chi tiết

+ Để chứng minh bốn điểm M, N, M’, N’ đồng phẳng ta có thể chứng minh hai đường thẳng MM’//NN’. + Tứ giác có 1 cặp cạnh đối song song và bằng nhau là hình bình hành. + Nếu hai mặt phẳng chứa hai đường thẳng song song với nhau thì giao tuyến của chúng (nếu có) song song với hai đường thẳng đó hoặc trùng với một trong hai đường thẳng đó.

Lời giải chi tiết

a) Vì M, M’ lần lượt là trung điểm của AB, A’B’ của hình bình hành ABB’A’ nên MM’//AA’ và \(MM' = AA'\)Tương tự ta có: NN’//DD’ và \(NN' = DD'\)Tứ giác ADD’A’ là hình bình hành nên AA’//DD’ và \(AA' = DD'\).Do đó, \(MM' = NN'\) và MM’//NN’, suy ra bốn điểm M, N, M’, N’ đồng phẳng và tứ giác MNN’M’ là hình bình hành.b) Trong mặt phẳng (ABCD), gọi P là giao điểm của hai đường thẳng MN và BC.Vì BB’// MM’ nên giao tuyến của hai mặt phẳng (MNN’M’) và (BCC’B’) là đường thẳng d qua P và song song với BB’.

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