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Giải bài 4.10 trang 51 sách bài tập toán 10 - Kết nối tri thức với cuộc sống

Cho tam giác ABC. Gọi D,E,F theo thứ tự là trung điểm của các cạnh BC,CA,AB.

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

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

Cho tam giác \(ABC.\) Gọi \(D,\,\,E,\,\,F\) theo thứ tự là trung điểm của các cạnh \(BC,\,\,CA,\,\,AB.\) a) Xác định vectơ \(\overrightarrow {AF}  - \overrightarrow {BD}  + \overrightarrow {CE} \) b) Xác định điểm \(M\) thỏa mãn \(\overrightarrow {AF}  - \overrightarrow {BD}  + \overrightarrow {CE}  = \overrightarrow {MA} .\) c) Chứng minh rằng \(\overrightarrow {MC}  = \overrightarrow {AB} .\)

Phương pháp giải - Xem chi tiết

- Chứng minh \(\overrightarrow {AF}  = \overrightarrow {FB} ,\) \(\overrightarrow {BD}  = \overrightarrow {DC} \) - Áp dụng quy tắc hình bình hành với hai vectơ \(\overrightarrow {CE} \) và \(\overrightarrow {CD} \) - Chứng minh tứ giác \(ABCM\) là hình bình hành

Lời giải chi tiết

a)      Ta có: \(DF\) là đường trung bình của \(\Delta ABC\)\( \Rightarrow \) \(\overrightarrow {CE}  = \overrightarrow {DF} \)\( \Rightarrow \) tứ giác \(CDFE\) là hình bình hành.Ta có: \(D\) và \(F\) lần lượt là trung điểm của \(BC\) và \(AB\)\( \Rightarrow \) \(\overrightarrow {AF}  = \overrightarrow {FB} ,\) \(\overrightarrow {BD}  = \overrightarrow {DC} \) Ta có: \(\overrightarrow {AF}  - \overrightarrow {BD}  + \overrightarrow {CE}  = \overrightarrow {AF}  + \overrightarrow {CD}  + \overrightarrow {CE}  = \overrightarrow {AF}  + \overrightarrow {CF}  = \overrightarrow {CF}  + \overrightarrow {FB}  = \overrightarrow {CB} \)b)     Theo câu a, ta có: \(\overrightarrow {AF}  - \overrightarrow {BD}  + \overrightarrow {CE}  = \overrightarrow {CB} \)mặt khác \(\overrightarrow {AF}  - \overrightarrow {BD}  + \overrightarrow {CE}  = \overrightarrow {MA} .\)nên \(\overrightarrow {CB}  = \overrightarrow {MA} \)\( \Rightarrow \) tứ giác \(ABCM\) là hình bình hành\( \Rightarrow \) \(M\) là điểm đối xứng với \(B\) qua \(E\)
c)      Theo câu b, ta có: tứ giác \(ABCM\) là hình bình hành\( \Rightarrow \) \(\overrightarrow {MC}  = \overrightarrow {AB} .\)

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