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Bài 10 trang 87 SGK Toán 11 tập 2 – Chân trời sáng tạo

Cho hình chóp \(S.ABCD\) có đáy là hình vuông cạnh \(a\), \(SA \bot \left( {ABCD} \right)\) và \(SA = a\).

Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn🦄 - Chân trời s𒀰áng tạo

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

Cho hình chóp \(S.ABCD\) có đáy là hình vuông cạnh \(a\), \(SA \bot \left( {ABCD} \right)\) và \(SA = a\). Gọi \(M,N,P\) lần lượt là trung điểm của \(SB,SC\) và \(SD\). Tính khoảng cách giữa \(AM\) và \(NP\).

Phương pháp giải - Xem chi tiết

Cách tính khoảng cách giữa hai đường thẳng chéo nhau:

Cách 1: Dựng đường vuông góc chung.

Cách 2: Tính khoảng cách từ đường thẳng này đến một mặt pꦇhẳng song song với đường thẳng đó và chứa đường thẳng còn lại.

Lời giải chi tiết

\(M\) là trung điểm của \(SB\)\(N\) là trung điểm của \(SC\)\( \Rightarrow MN\) là đường trung bình của \(\Delta SBC\)\(\left. \begin{array}{l} \Rightarrow MN\parallel BC\\BC \bot C{\rm{D}}\end{array} \right\} \Rightarrow MN \bot C{\rm{D}}\)Mà \(C{\rm{D}}\parallel NP\) \( \Rightarrow MN \bot NP\) (1)\(\left. \begin{array}{l}SA \bot \left( {ABCD} \right) \Rightarrow SA \bot BC\\AB \bot BC\end{array} \right\} \Rightarrow BC \bot \left( {SAB} \right)\)Mà \(MN\parallel BC\)\( \Rightarrow MN \bot \left( {SAB} \right) \Rightarrow MN \bot AM\)(2)Từ (1) và (2) \( \Rightarrow d\left( {AM,NP} \right) = MN = \frac{1}{2}BC = \frac{a}{2}\).

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