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Bài tập 4 trang 156 Tài liệu dạy – học Toán 7 tập 1

Giải bài tập Cho tam giác ABC có AB = AC. Gọi M là trung điểm của BC.
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Đề bài

Cho tam giác ABC có AB = AC. Gọi M là trung điểm của BC. a) Chứng minh rằng AM là tia phân giác của góc A. b) Kẻ ME vuông góc với AB, MF vuông góc với AC. Chứng minh rằng AE = AF.

Lời giải chi tiết

 

a)Xét tam giác ABM và ACM có:AB = AC (gt)BM = CM (M là trung điểm của BC)AM là cạnh chung.Do đó: \(\Delta ABM = \Delta ACM(c.c.c) \Rightarrow \widehat {BAM} = \widehat {CAM}\)Vậy AM là tia phân giác của góc BAC.b) Xét hai tam giác vuông EBM và FCM có:BM = CM (M là trung điểm của BC)\(\widehat {EBM} = \widehat {FCM}(do\Delta ABM = \Delta ACM)\)Do đó: \(\Delta EBM = \Delta FCM\)  (cạnh huyền - góc nhọn) => BE = CF.Ta có: AE + BE = AB và AF + CF = ACMà AB = AC (giả thiết) và BE = CF (chứng minh trên) nên AE = AF.

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