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Bài 8.2 trang 54 SGK Toán 11 tập 2 - Cùng khám phá

Cho hình hộp \(ABCD.A'B'C'D'\) có tất cả các cạnh đều bằng nhau. Chứng minh rằng \(AC \bot B'D'\), \(AB' \bot CD'\) và \(AD' \bot CB'\)
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Đề bài

Cho hình hộp \(ABCD.A'B'C'D'\) có tất cả các cạnh đều bằng nhau. Chứng minh rằng \(AC \bot B'D'\), \(AB' \bot CD'\) và \(AD' \bot CB'\)

Phương pháp giải - Xem chi tiết

Góc giữa hai đường thẳng \(a,b\) là góc giữa hai đường thẳng \(a',b'\) cùng đi qua một điểm và lần lượt song song hoặc trùng với \(a,b\). Dựa vào tính chất: hình thoi có hai đường chéo vuông góc với nhau

Lời giải chi tiết

+) Vì \(B'D'//BD\)\( \Rightarrow \left( {AC,B'D'} \right) = \left( {AC,BD} \right)\)Vì \(ABCD\) là hình thoi nên \(AC \bot BD\)\( \Rightarrow \left( {AC,BD} \right) = {90^o}\)\( \Rightarrow \left( {AC,B'D'} \right) = {90^o} \Rightarrow AC \bot B'D'\)+) Vì \(AB'//DC' \Rightarrow \left( {AB',CD'} \right) = \left( {DC',CD'} \right)\)Vì \(CDD'C'\) là hình thoi nên \(CD' \bot DC'\)\( \Rightarrow \left( {DC',CD'} \right) = {90^o}\)\( \Rightarrow \left( {AB',CD'} \right) = {90^o} \Rightarrow AB' \bot CD'\)+) Vì \(A'D//B'C \Rightarrow \left( {AD',CB'} \right) = \left( {AD',A'D} \right)\)Vì \(ADD'A'\) là hình thoi nên \(A'D \bot AD'\)\( \Rightarrow \left( {AD',CB'} \right) = {90^o} \Rightarrow AD' \bot CB'\)

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