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Giải bài 6 trang 76 SGK Toán 7 tập 2 - Chân trời sáng tạo

Cho tam giác ABC cân tại A có BD và CE là hai đường trung tuyến cắt nhau tại F (Hình 10). Biết BE = 9 cm, tính độ dài đoạn thẳng DF.

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Đề bài

Cho tam giác ABC cân tại A có BD và CE là hai đường trung tuyến cắt nhau tại F (Hình 10). Biết BE = 9 cm, tính độ dài đoạn thẳng DF.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

- Ta chứng minh F là trọng tâm tam giác ABC - Sau đó chứng minh CD = BE - Áp dụng định lí về trọng tâm tam giác ta tính các đoạn DF, EF

Lời giải chi tiết

Vì BE, CD là 2 trung tuyến của tam giác ABC nên E, D lần lượt là trung tuyến của AB và AC\( \Rightarrow AD = AE = \dfrac{1}{2}AB = \dfrac{1}{2}AC\)Xét tam giác ADC và tam giác AEB có :AD = AE (gt)\(\widehat{A}\) chungAB = AC (do \(\Delta ABC\) cân tại A )\( \Rightarrow \Delta ADC = \Delta AEB(c - g - c)\)\( \Rightarrow BE = CD\)(cạnh tương ứng)Tam giác ABC có F là giao điểm của 2 trung tuyến BE, CD nên F là trọng tâm tam giác ABC\( \Rightarrow CF = BF = \dfrac{2}{3}BE = \dfrac{2}{3}CD\) ( định lí về trung tuyến đi qua trọng tâm tam giác )\( \Rightarrow \dfrac{1}{3}BE = \dfrac{1}{3}CD \Rightarrow DF = FE = \dfrac{1}{3}.9cm = 3cm\)\( \Rightarrow \) DF = 3 cm 

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