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Bài tập 6 trang 120 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC có trung tuyến BM bằng trung tuyến CN. Chứng minh tam giác ABC cân.
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Đề bài

Cho tam giác ABC có trung tuyến BM bằng trung tuyến CN. Chứng minh tam giác ABC cân.

Lời giải chi tiết

 

Gọi I là giao điểm của BM và CN.∆ABC có hai đường trung tuyến BM và CN cắt nhau tại I=> I là trọng tâm của tam giác ABC \( \Rightarrow CI = {2 \over 3}CN\) và\(BI = {2 \over 3}BM\)Mà CN = BM (gt). Nên CI = BI => ∆BIC cân tại I.Xét ∆NCB và ∆MBC ta có: NC = BM (gt)\(\widehat {NCB} = \widehat {MBC}\) (∆IBC cân tại I)BC là cạnh chungDo đó ∆NCB = ∆MBC (c.g.c)Suy ra \(\widehat {NBC} = \widehat {MCB}\,hay,\widehat {ABC} = \widehat {ACB}\)Vậy ∆ABC cân tại A.

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