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Bài 8.48 trang 90 SGK Toán 11 tập 2 - Cùng khám phá

Cho hình chóp S.ABCD có đáy ABCD là hình chữ nhật, AB = a, BC = 2a, SA vuông góc với mặt phẳng đáy và SA = a.
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là hình chữ nhật, AB = a, BC = 2a, SA vuông góc với mặt phẳng đáy và SA = a. Khoảng cách giữa hai đường thẳng BD, SC bằng A. \(\frac{{\sqrt {30} }}{6}a\) B. \(\frac{{4\sqrt {21} }}{{21}}a\) C. \(\frac{{2\sqrt {21} }}{{21}}a\) D. \(\frac{{\sqrt {30} }}{{12}}a\)

Phương pháp giải - Xem chi tiết

Tìm mặt phẳng chứa BD và song song với SC. Sau đó tính khoảng cách.

Lời giải chi tiết

Gọi O là trung điểm BD, M là trung điểm SASC // (BMD)Nên \(d\left( {SC,BD} \right) = d\left( {SC,\left( {BMD} \right)} \right) = d\left( {S,\left( {BMD} \right)} \right) = d\left( {A,\left( {BMD} \right)} \right) = h\)\(\frac{1}{{{h^2}}} = \frac{1}{{A{M^2}}} + \frac{1}{{A{B^2}}} + \frac{1}{{A{D^2}}} = \frac{1}{{\frac{1}{4}{a^2}}} + \frac{1}{{{a^2}}} + \frac{1}{{4{a^2}}}\)\( \Rightarrow h = \frac{{2\sqrt {21} a}}{{21}}\)Chọn đáp án C.

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