Cho hình chóp \(\mathrm{S} . \mathrm{ABC}\) có đáy ABC là tam giác vuông tại \(\mathrm{B}, \mathrm{SA}\) vuông góc với mặt đáy và \(S A=A B=\sqrt{3}\). Gọi \(G\) là trọng tâm của tam giác SAB . Khoảng cách từ \(G\) đến mặt phẳng \((S B C)\) bằng:
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Phương pháp giải
Gọi \(M\) là trung điểm của SB . Chứng minh \(G M \perp(S B C)\).
Khi đó, \(d(G ;(S B C))=G M\).
Tính khoảng cách từ một điểm đến một mặt phẳng.
Lời giải
Gọi \(M\) là trung điểm của \(S B \Rightarrow A M \perp S B\) (vì \(\triangle S A B\) cân)
Ta có: \(\left\{\begin{array}{l}B C \perp A B B C \perp S A\end{array} \Rightarrow B C \perp(S A B) \Rightarrow B C \perp A M\right.\)
Và \(\left\{\begin{array}{l}A M \perp S B A M \perp B C\end{array} \Rightarrow A M \perp(S B C) \Rightarrow G M \perp(S B C)\right.\) tại \(M\).
Do đó \(d(G ;(S B C))=G M\).
Ta có: \(S M=\sqrt{A B^{2}+S A^{2}}=\sqrt{6} \Rightarrow A M=\frac{S B}{2}=\frac{\sqrt{6}}{2}\).
Vì \(G\) là trọng tâm của \(\triangle S A B\) nên \(G M=\frac{1}{3} A M=\frac{\sqrt{6}}{6}\).
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