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Câu 2 trang 91 SGK Hình học 11 Nâng cao

Cho hình chóp S.ABCD.
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Đề bài

Cho hình chóp S.ABCD. a. Chứng minh rằng nếu ABCD là hình bình hành thì \(\overrightarrow {SB}  + \overrightarrow {SD}  = \overrightarrow {SA}  + \overrightarrow {SC} \). Điều ngược lại có đúng không ? b. Gọi O là giao điểm của AC và BD. Chứng tỏ rằng ABCD là hình bình hành khi và chỉ khi \(\overrightarrow {SA}  + \overrightarrow {SB}  + \overrightarrow {SC}  + \overrightarrow {SD}  = 4\overrightarrow {SO} \)

Lời giải chi tiết

 

a. Ta có:\(\eqalign{  & \overrightarrow {SB}  + \overrightarrow {SD}  = \overrightarrow {SA}  + \overrightarrow {SC}   \cr  &  \Leftrightarrow \overrightarrow {SB}  - \overrightarrow {SC}  = \overrightarrow {SA}  - \overrightarrow {SD}  \cr&\Leftrightarrow \overrightarrow {CB}  = \overrightarrow {DA}  \cr} \)⇔ ABCD là hình bình hành.b. Ta có:\(\eqalign{  & \overrightarrow {SA}  + \overrightarrow {SB}  + \overrightarrow {SC}  + \overrightarrow {SD}  = 4\overrightarrow {SO}   \cr  &  \Leftrightarrow \overrightarrow {SO}  + \overrightarrow {OA}  + \overrightarrow {SO}  + \overrightarrow {OB} \cr& + \overrightarrow {SO}  + \overrightarrow {OC}  + \overrightarrow {SO}  + \overrightarrow {OD}  = 4\overrightarrow {SO}   \cr  &  \Leftrightarrow \overrightarrow {OA}  + \overrightarrow {OB}  + \overrightarrow {OC}  + \overrightarrow {OD}  = \overrightarrow 0 \,\,\left( * \right) \cr} \)Nếu ABCD là hình bình hành thì \(\overrightarrow {OA}  + \overrightarrow {OB}  + \overrightarrow {OC}  + \overrightarrow {OD}  = \overrightarrow 0 \) suy ra
 \(\overrightarrow {SA}  + \overrightarrow {SB}  + \overrightarrow {SC}  + \overrightarrow {SD}  = 4\overrightarrow {SO} \) (do (*))Ngược lại, giả sử \(\overrightarrow {SA}  + \overrightarrow {SB}  + \overrightarrow {SC}  + \overrightarrow {SD}  = 4\overrightarrow {SO} ,\) ta có (*).Gọi M, N lần lượt là trung điểm của AC, BD thì :\(\overrightarrow {OA}  + \overrightarrow {OC}  = 2\overrightarrow {OM} ,\overrightarrow {OB}  + \overrightarrow {OD}  = 2\overrightarrow {ON} \)Từ (*) suy ra \(2\left( {\overrightarrow {OM}  + \overrightarrow {ON} } \right) = \overrightarrow 0 ,\) điều này chứng tỏ O, M, N thẳng hàngMặt khác, M thuộc AC, N thuộc BD và O là giao điểm của AC và BD nên O, M, N thẳng hàng chỉ xảy ra khi O ≡ M ≡ N, tức O là trung điểm AC và BD, hay ABCD là hình bình hành.

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