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Giải bài 5 trang 76 SGK Toán 7 tập 2 - Chân trời sáng tạo

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

Tổng hợp đề thi học🍬 kì 2 l⛄ớp 7 tất cả các môn - Chân trời sáng tạo

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Đề bài

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

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

- Ta chứng minh AB = AC bằng cách chứng minh 2 tam giác bằng nhau

Lời giải chi tiết

Gọi D là giao điểm của CN và BM \( \Rightarrow \) D là trọng tâm tam giác ABC \( \Rightarrow CD = \dfrac{2}{3}CN = BD = \dfrac{2}{3}BM\) ( do BM = CN ) \( \Rightarrow \) tam giác DBC cân tại D do BD = CD \( \Rightarrow \) \(\widehat {DBC} = \widehat {DCB}\)(2 góc đáy trong tam giác cân)  (1) Xét \(\Delta NDB\) và \(\Delta MDC\) có : BD = CD \(\widehat {NDB} = \widehat {MDC}\) (2 góc đối đỉnh) ND = DM (do cùng \( = \dfrac{1}{3}CN = \dfrac{1}{3}BM\) (tính chất của trung trực đi qua trọng tâm tam giác ))  \( \Rightarrow \Delta NDB=\Delta MDC\) (c.g.c) \( \Rightarrow \,\widehat {NBD} = \widehat {MCD}\)(2 góc tương ứng) (2) Từ (1) và (2) \( \Rightarrow \widehat {ABC} = \widehat {ACB}\) do \(\widehat {ABC} = \widehat {NBD} + \widehat {DBC}\) và \(\widehat {ACB} = \widehat {MCD} + \widehat {DCB}\) \( \Rightarrow \Delta ABC\) cân tại A (do 2 góc bằng nhau)

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