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Giải bài 4.38 trang 68 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hình hộp ABCD.A’B’C’D’. Gọi O là giao điểm của các đường chéo của hình hộp.

ཧTổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nối tri thức

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

Cho hình hộp ABCD.A’B’C’D’. Gọi O là giao điểm của các đường chéo của hình hộp. Mặt phẳng qua O và song song với mặt phẳng (ABCD) cắt các cạnh AA’, BB’, CC’, DD’ lần lượt tại M, N, P, Q. a) Chứng minh rằng M, N, P, Q lần lượt là trung điểm của các cạnh AA’, BB’, CC’, DD’. b) Chứng minh rằng ABCD.MNPQ là hình hộp.

Phương pháp giải - Xem chi tiết

Định lí Thalès: Ba mặt phẳng đôi một song song chắn trên hai cát tuyến phân biệt bất kì những đoạn thẳng tương ứng tỉ lệ Hình lăng trụ tứ giác ABCD.A’B’C’D’ có hai đáy là hình bình hành được gọi là hình hộp.

Lời giải chi tiết

a) Áp dụng định lí Thalès cho ba mặt phẳng (ABCD), (MNPQ), (A’B’C’D’) và hai cát tuyến AA’, DB’ ta có: \(\frac{{AM}}{{MA'}} = \frac{{DO}}{{OB'}}\)Vì O là trung điểm của DB’ nên M là trung điểm của AA’.Chứng minh tương tự ta có: N, P, Q lần lượt là trung điểm của BB’, CC’, DD’.b) Vì M, N lần lượt là trung điểm của AA’, BB’ nên MN//AB, \(MN = AB\)Tương tự ta có: PQ//CD và \(PQ = CD\)Vì \(AB = CD\) và AB//CD nên \(MN = PQ\) và MN//PQ.Do đó tứ giác MNPQ là hình bình hành.Vì các đường thẳng AM, BN, CP, DQ đôi một song song nên suy ra ABCD.MNPQ là hình hộp.

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