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Giải bài 3 trang 75 SGK Toán 7 tập 2 - Chân trời sáng tạo

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

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Đề bài

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

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

- Ta dựa vào định lí ba đường trung tuyến cắt nhau tại 1 điểm. Điểm đó cách mỗi đỉnh một khoảng bằng \(\dfrac{2}{3}\)độ dài trung tuyến đi qua đỉnh ấy - Câu a ta sẽ chứng minh 2 góc so le trong bằng nhau thông qua các tam giác bằng nhau - Câu b ta sẽ chứng minh F là trọng tâm tam giác ABE

Lời giải chi tiết

a) Xét tam giác BGM và tam giác CEM có :\(\widehat {GMB} = \widehat {EMC}\)(2 góc đối đỉnh)GM = ME (do G đối xứng E qua M)MB = MC (do M là trung điểm của BC)\( \Rightarrow \Delta BGM = \Delta CEM(c - g - c)\)\( \Rightarrow \widehat {GBM} = \widehat {MCE}\)(2 góc tương ứng bằng nhau)Mà 2 góc trên ở vị trí so le trong nên BG⫽CEb) Vì I là trung điểm BE nên AI sẽ là trung tuyến của tam giác ABEVà BG cũng là trung tuyến của tam giác ABE do G là trung điểm AEVì BG cắt AI tại F nên F sẽ là trọng tâm của tam giác ABE\(\, \Rightarrow AF = \dfrac{2}{3}AI\)(định lí về trọng tâm tam giác)Mà AI = AF + FI \( \Rightarrow \) FI = AI – AF\( \Rightarrow FI = AI - \dfrac{2}{3}AI = \dfrac{1}{3}AI\)\( \Rightarrow 2FI = AF = \dfrac{2}{3}AI\)\( \Rightarrow \) AF = 2 FI

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