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Giải Bài 9.15 trang 55 sách bài tập toán 7 - Kết nối tri thức với cuộc sống

Gọi M là trung điểm của cạnh BC của tam giác ABC và D là điểm sao cho M là trung điểm của AD. Đường thẳng qua D và trung điểm của AB cắt BC tại U, đường thẳng qua D và trung điểm của AC cắt BC tại V. Chứng minh BU = UV = VC.

🦩Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Kết nối tri thức

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Đề bài

Gọi M là trung điểm của cạnh BC của tam giác ABC và D là điểm sao cho M là trung điểm của AD. Đường thẳng qua D và trung điểm của AB cắt BC tại U, đường thẳng qua D và trung điểm của AC cắt BC tại V. Chứng minh BU = UV = VC.

Phương pháp giải - Xem chi tiết

-Chứng minh: U là trọng tâm tam giác ABD. -Chứng minh: V là trọng tâm tam giác ACD -MB = MC

Lời giải chi tiết

-Xét tam giác ABD có:M là trung điểm của AD, DU đi qua trung điểm AB=>BM và DU là 2 đường trung tuyến của tam giácMà BM cắt DU tại U=>U là trọng tâm tam giác ABD.\( \Rightarrow BU = 2UM = \dfrac{2}{3}BM\)(1)-Xét tam giác ACD:M là trung điểm của AD, DV đi qua trung điểm AC=>CM và DV là 2 đường trung tuyến của tam giácMà CM cắt DV tại V=>V là trọng tâm tam giác ACD.\( \Rightarrow CV = 2MV = \dfrac{2}{3}MC\)(2)Mà M là trung điểm BC\( \Rightarrow MB = MC\)Lại có: UV = UM + MV = \(\dfrac{1}{3}BM + \dfrac{1}{3}CM = \dfrac{1}{3}BM + \dfrac{1}{3}BM = \dfrac{2}{3}BM\) (3)Từ (1), (2) và (3) suy ra:BU = UV = VC. 

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