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Giải bài 24 trang 69 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hình chóp \(S.ABC\) có đáy \(ABC\) là tam giác đều cạnh bằng \(a\) và \(SA\) vuông góc với mặt phẳng \(\left( {ABC} \right)\).

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Đề bài

Cho hình chóp \(S.ABC\) có đáy \(ABC\) là tam giác đều cạnh bằng \(a\) và \(SA\) vuông góc với mặt phẳng \(\left( {ABC} \right)\). Khoảng cách giữa hai đường thẳng \(SA\) và \(BC\) bằng

A. \(\frac{{a\sqrt 2 }}{2}\).

B. \(\frac{{a\sqrt 3 }}{4}\).

C. \(\frac{{a\sqrt 6 }}{2}\).

D. \(\frac{{a\sqrt 3 }}{2}\).

Phương pháp giải - Xem chi tiết

Tìm đoạn vuông góc chung của \(SA\) và \(BC\) Gọi \(M\) là trung điểm \(BC\) Chứng minh \(AM\) là đoạn vuông góc chung của \(SA\) và \(BC\) Khoảng cách giữa hai đường thẳng \(SA\) và \(BC\) bằng \(AM\)

Lời giải chi tiết

Gọi \(M\) là trung điểm \(BC \Rightarrow AM \bot BC\) do áy \(ABC\) là tam giác đềuTa có \(SA \bot \left( {BC} \right) \Rightarrow AM \bot SA\)Khoảng cách giữa hai đường thẳng \(SA\) và \(BC\) bằng \(AM = \frac{{a\sqrt 3 }}{2}\)

Chọn D

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