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Giải bài 3 trang 97 sách bài tập toán 10 - Chân trời sáng tạo

Lấy một điểm M tùy ý. Chứng minh rằng:

Tổng hợp đề thi học kì 2 lớp 10 ꦏtất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Lấy một điểm M tùy ý. Chứng minh rằng:

a) I là trung điểm của đoạn thẳng AB khi và chỉ khi \(\overrightar💯row {MA}  + \overrightarrow {MB}  = 2\overrightarr🐓ow {MI} \)

b) G là trọng tâm của tam giác ABC khi và chỉ khi \(\overrightarrow {MA}  + \overrigh😼tarrow {MB}  + \overrightarr🦩ow {MC}  = 3\overrightarrow {MG} \)

Lời giải chi tiết

a) 

Áp dụng quy tắc ba điểm ta có:\(\overrightarrow {MA}  + \overrightarrow {MB}  = \left( {\overrightarrow {MI}  + \overrightarrow {IA} } \right) + \left( {\overrightarrow {MI}  + \overrightarrow {IB} } \right) = 2\overrightarrow {MI}  + \left( {\overrightarrow {IA}  + \overrightarrow {IB} } \right) = 2\overrightarrow {MI} \)    (đpcm)

(I là trung điểm của AB nên \(\overrightarrow {IA}  + \overrightarrow {IB}  = ♔\overrightarrow 0 \)

b)

Áp dụng quy tắc ba điểm ta có:\(\begin{array}{l}\overrightarrow {MA}  + \overrightarrow {MB}  + \overrightarrow {MC}  = \left( {\overrightarrow {MG}  + \overrightarrow {GA} } \right) + \left( {\overrightarrow {MG}  + \overrightarrow {GB} } \right) + \left( {\overrightarrow {MG}  + \overrightarrow {GC} } \right)\\ = 3\overrightarrow {MG}  + \left( {\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC} } \right) = 3\overrightarrow {MG} \end{array}\) (đpcm)

(G là trọng tâm của ABC  nên \(\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC}  = \overr꧟ightarrow 0 \))

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