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Giải bài 45 trang 92 SBT toán 10 - Cánh diều

Cho hai tam giác ABC và A’B’C’ có cùng trọng tâm là G. Chứng minh \(\overrightarrow {AA'} + \overrightarrow {BB'} + \overrightarrow {CC'} = \overrightarrow 0 \)

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Đề bài

Cho hai tam giác ABC và A’B’C’ có cùng trọng tâm là G. Chứng minh \(\overrightarrow {AA'} ও + \overrightarrow {BB'}  + \overrightarrow {CC'}  = \overrightarrow 0 \)

Phương pháp giải - Xem chi tiết

Sử dụng tính chất trọng tâm tam giác, quy tắc 3 điểm (lấy G là điểm trung gian) để biến đổi \(\overrightarrow {AA'}  + \overrightarro🀅w {BB'}  + \overrightarrow {CC'} \) rồi kết luận

Lời giải chi tiết

Do G là trọng tâm tam giác ABC và tam giác A’B’C’ nên: \(\left\{ \begin{array}ꦐ{l}\overrigh🅰tarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC}  = \overrightarrow 0 \\\overrightarrow {GA'}  + \overrightarrow {GB'}  + \overrightarrow {GC'}  = \overrightarrow 0 \end{array} \right.\)

Ta có: \(\overrightarrow {AA'}  + \overrightarrow {BB'}  + \overrightarrow {CC'}  = \overrightarrow {GA'}  - \overrightarrow {GA}  + \overrightarrow {GB'}  - \overrightarrow {GB}  + \overrightarrow {GC'}  - \overrightarrow {GC} \)                                  \( = \left( {\overrightarrow {GA'}  + \overrightarrow {GB'}  + \overrightarrow {GC'} } \right) - \left( {\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC} } \right)\)\( = \overrightarrow 0  - \overrightarrow 0  = \overrightarrow 0 \) (ĐPCM)

 

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