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Bài tập 3 trang 120 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác nhọn ABC. Trung tuyến AM và CN cắt nhau tại G. Trên tia đối của tia MA lấy điểm E sao cho ME = MG.

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Đề bài

Cho tam giác nhọn ABC. Trung tuyến AM và CN cắt nhau tại G. Trên tia đối của tia MA lấy điểm E sao cho ME = MG. a) Chứng minh BG song song với EC. b) Gọi I là trung điểm của BE, AI cắt BG tại F. Chứng minh : AF = 2FI.

Lời giải chi tiết

 

a) Xét ∆BMG và ∆CME ta cóBM = MC (M là trung điểm của BC)\(\widehat {BMG} = \widehat {CME}\) (hai góc đối đỉnh)Và GM = ME (gt)Do đó: ∆BMG = ∆CME (c.g.c) \( \Rightarrow \widehat {BGM} = \widehat {CEM}\)Mà \(\widehat {BGM}\) và \(\widehat {CEM}\) ở vị trí so le trong nên BG // EC.b) ∆ABC có hai đường trung tuyến AM và CN cắt nhau tại G (gt)=> G là trọng tâm của ∆ABC \( \Rightarrow AG = {2 \over 3}AM\)Mà AG + GM = AM. Do đó \(GM = {1 \over 3}AM.\) Nên AG = 2GM.Mà MG = ME => AG = GE. Vậy G là trung điểm của AE.∆ABE có: BG và AI cắt nhau tại F (gt)AI là đường trung tuyến (I là trung điểm của BE)Và BG là đường trung tuyến (G là trung điểm của AE)Do đó F là trọng tâm của tam giác ABE \( \Rightarrow AF = {2 \over 3}AI\)Mà AF + FI = AI; \(FI = AI - {2 \over 3}AI = {1 \over 3}AI.\) Nên AF = 2FI.

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