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Giải bài 7.44 trang 42 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho tứ diện đều \(ABCD\)có cạnh bằng\(a\), côsin của góc giữa đường thẳng \(AB\) và mặt phẳng \(\left( {BCD} \right)\)bằng

🍌Tổng 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 tứ diện đều \(ABCD\)có cạnh bằng\(a\), côsin của góc giữa đường thẳng \(AB\) và mặt phẳng \(\left( {BCD} \right)\)bằng

A. \(\frac{1}{3}\).

B. \(\frac{{\sqrt 3 }}{3}\).

C. \(\frac{{\sqrt 3 }}{2}\).

D. \(\frac{1}{2}\).

Phương pháp giải - Xem chi tiết

- Chóp có các cạnh bên bằng nhau có chân đường cao trùng với tâm đường tròn ngoại tiếp đáy. - Góc giữa đường và mặt là góc giữa đường thẳng và hình chiếu của đường thẳng trên mặt phẳng. - Sử dụng tỉ số lượng giác của góc nhọn trong tam giác vuông.

Lời giải chi tiết

 🐽Gọi \(M\)là trung điểmcủa \(CD,O\) là tâm đường tròn ngoại tiếp tam giác đều \(BCD\)⇒\(AO \bot (BCD)\)

Khi đó \(OB\)là hình chiếu vuông góc của \(AB\) lên \((BCD)\)\( \Rightarrow (AB;(BCD)) = (AB;OB) = \widehat {ABO}\)Tam giác \(BCD\) đều cạnh a nên \(BM = \frac{{a\sqrt 3 }}{2} \Rightarrow BO = \frac{{2BM}}{3} = \frac{{a\sqrt 3 }}{3}\).Ta có \(AO \bot (BCD)\) nên\(AO \bot OB\), suy ra \(\Delta ABO\)vuông tại \(O\).⇒\(cos\widehat {ABO} = \frac{{OB}}{{AB}} = \frac{{\sqrt 3 }}{3}\) Vậy \(\cos (AB;(BCD)) = \frac{{\sqrt 3 }}{3}\)

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