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Giải bài 7.13 trang 30 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho tứ diện \(ABCD\) có tất cả các cạnh bằng nhau và bằng \(a\).

Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết n♒ối tri thức

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

Cho tứ diện \(ABCD\) có tất cả các cạnh bằng nhau và bằng \(a\). Tính côsin của góc giữa đường thẳng \(AB\) và mặt phẳng \(BCD\).

Phương pháp giải - Xem chi tiết

Phương pháp chung Để xác định góc giữa đường thẳng \(a\) và mặt phẳng \(\left( \alpha  \right)\)ta thực hiện theo các bước sau: -   Tìm giao điểm \(O = a \cap \left( \alpha  \right)\) -    Dựng hình chiếu \(A'\) của một điểm \(A \in a\)  xuống \(\left( \alpha  \right)\) -    Góc \(\widehat {AOA'} = \varphi \) chính là góc giữa đường thẳng \(a\) và \(\left( \alpha  \right)\).

Gợi ý phương pháp giải

Kẻ \(AH \bot \left( {BCD} \right)\) tại \(H\), Xác định hình chiếu của \(AB\) trên \(\left( {BCD} \right)\) là \(BH\) Tính góc \(\left( {AB,BH} \right) = \widehat {ABH}\) rồi kết luận

Lời giải chi tiết

Kẻ \(AH \bot \left( {BCD} \right)\) tại \(H\), ta có \(BH\) là hình chiếu vuông góc của \(AB\) trên mặt phẳng \(\left( {BCD} \right)\) nên góc giữa đường thẳng \(AB\) và mặt phẳng \(\left( {BCD} \right)\) bằng góc giữa hai đường thẳng \(AB\) và \(BH\), mà \(\left( {AB,BH} \right) = \widehat {ABH}\).Vì \(AB = AC = AD\)  nên \(HB = HC = HD\), hay \(H\)  là tâm của tam giác\(BCD\), suy ra\(BH = \frac{2}{3}.\frac{{a\sqrt 3 }}{2} = \frac{{a\sqrt 3 }}{3}\).Từ đó ta tính được:  \(\cos \widehat {ABH} = \frac{{BH}}{{AB}} = \frac{{\sqrt 3 }}{3}\).

 

Vậy côsin của góc giữa đường thẳng \(AB\) và mặt phẳng \(\left( {BCD} \right)\) bằng \(\frac{{\sqrt 3 }}{3}\).

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