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Giải bài 4.26 trang 70 SGK Toán 10 – Kết nối tri thức

Cho tam giác ABC có trọng tâm G. Chứng minh rằng với mọi điểm M, ta có

Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Kết nốiꦚ tri thức

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Đề bài

Cho tam giác ABC có trọng tâm G. Chứng minh rằng với mọi điểm M, ta có: \(M{A^2} + M{B^2} + M{C^2} = 3M{G^2} + G{A^2} + G{B^2} + G{C^2}\)

Phương pháp giải - Xem chi tiết

+) \(M{A^2} = {\overrightarrow {MA} ^2}\) +) Với 3 điểm M, A, G bất kì ta có: \(\overrightarrow {MG}  + \overrightarrow {GA}  = \overrightarrow {MA} \) +) G là trọng tâm tam giác ABC thì: \(\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC}  = \overrightarrow 0 \)

Lời giải chi tiết

Ta có: \(\begin{array}{l}M{A^2} + M{B^2} + M{C^2} = {\overrightarrow {MA} ^2} + {\overrightarrow {MB} ^2} + {\overrightarrow {MC} ^2}\\ = {\left( {\overrightarrow {MG}  + \overrightarrow {GA} } \right)^2} + {\left( {\overrightarrow {MG}  + \overrightarrow {GB} } \right)^2} + {\left( {\overrightarrow {MG}  + \overrightarrow {GC} } \right)^2}\\ = {\overrightarrow {MG} ^2} + 2\overrightarrow {MG} .\overrightarrow {GA}  + {\overrightarrow {GA} ^2} + {\overrightarrow {MG} ^2} + 2\overrightarrow {MG} .\overrightarrow {GB}  + {\overrightarrow {GB} ^2} + {\overrightarrow {MG} ^2} + 2\overrightarrow {MG} .\overrightarrow {GC}  + {\overrightarrow {GC} ^2}\\ = 3{\overrightarrow {MG} ^2} + 2\overrightarrow {MG} .\left( {\overrightarrow {GA}  + \overrightarrow {GB}  + \overrightarrow {GC} } \right) + {\overrightarrow {GA} ^2} + {\overrightarrow {GB} ^2} + {\overrightarrow {GC} ^2}\\ = 3{\overrightarrow {MG} ^2} + 2\overrightarrow {MG} .\overrightarrow 0  + {\overrightarrow {GA} ^2} + {\overrightarrow {GB} ^2} + {\overrightarrow {GC} ^2}\end{array}\)
( do G là trọng tâm tam giác ABC)\(\begin{array}{l} = 3{\overrightarrow {MG} ^2} + {\overrightarrow {GA} ^2} + {\overrightarrow {GB} ^2} + {\overrightarrow {GC} ^2}\\ = 3M{G^2} + G{A^2} + G{B^2} + G{C^2}\end{array}\) (đpcm).

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