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Giải Bài 76 trang 90 sách bài tập toán 7 - Cánh diều

Cho tam giác nhọn ABC. Trên tia đối của tia AB lấy điểm D sao cho AD = AB. Trên cạnh AC lấy điểm E sao cho \(A{\rm{E}} = \frac{1}{3}AC\).

🦹Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Cánh diều

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Đề bài

 🅺Cho tam giác nhọn ABC. Trên tia đối của tia AB lấy điểm D sao cho AD = AB. Trên cạnh AC lấy điểm E sao cho \(A{\rm{E}} = \frac{1}{3}AC\).

a) Chứng minh E là trọng tâm tam giác BCD. b) Gọi M là trung điểm DC. Chứng minh ba điểm B, M, E thẳng hàng.

Lời giải chi tiết

 

a) Ta có \(A{\rm{E}} = \frac{1}{3}AC\)nên \(CE = \frac{2}{3}AC\)Trong tam giác BCD có CA là trung tuyến và \(CE = \frac{2}{3}AC\).Suy ra E là trọng tâm tam giác BCD.Vậy E là trọng tâm tam giác BCD.b) Trong tam giác BCD có CA và BM là hai đường trung tuyến nên BM cắt CA tại trọng tâm của tam giác.Mà E là trọng tâm của tam giác BCD (theo câu a) nên điểm E thuộc đường thẳng BM.Hay ba điểm B, E, M thẳng hàng.Vậy ba điểm B, E, M thẳng hàng.

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