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Bài 8.3 trang 54 SGK Toán 11 tập 2 - Cùng khám phá

Cho hình chóp \(S.ABCD\) có đáy là hình vuông \(ABCD\) cạnh bằng \(a\)
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Đề bài

Cho hình chóp \(S.ABCD\) có đáy là hình vuông \(ABCD\) cạnh bằng \(a\) và các cạnh bên đều bằng \(a\). Gọi \(M\) và \(N\) lần lượt là trung điểm của \(AD\) và \(SD\). Chứng minh \(MN \bot SC\).

Phương pháp giải - Xem chi tiết

Chứng minh \(MN//SA\) Chứng minh \(SA \bot SC\) dựa vào việc tính các cạnh của tam giác \(SAC\)

Lời giải chi tiết

 Vì \(MN//SA\) (tính chất đường trung bình của tam giác)Do đó, \(\left( {MN,SC} \right) = \left( {SA,SC} \right) = \widehat {CSA}\)Vì \(ABCD\) là hình vuông cạnh \(a\) nên \(AC = a\sqrt 2 \).Xét \(\Delta SAC\) có \(SA = SC = a,AC = a\sqrt 2 \)\( \Rightarrow S{A^2} + S{C^2} = {a^2} + {a^2} = 2{a^2} = {\left( {\sqrt 2 a} \right)^2} = A{C^2}\)\( \Rightarrow \Delta SAC\) vuông tại \(S\) (theo định lí Pi-ta-go)\( \Rightarrow SA \bot SC \Rightarrow MN \bot SC\)

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