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Giải Bài 13 trang 70 sách bài tập toán 7 - Cánh diều

Cho tam giác ABC có góc A tù. Trên cạnh AC lấy điểm D và E (D nằm giữa A và E). Chứng minh BA < BD < BE < BC.

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Đề bài

Cho tam giác ABC có góc A tù. Trên cạnh AC lấy điểm D và E (D nằm giữa A và E). Chứng minh BA < BD < BE < BC.

Phương pháp giải - Xem chi tiết

Áp dụng mối quan hệ giữa góc và cạnh đối diện để chứng minh BA < BD < BE < BC.

Lời giải chi tiết

 

• Xét tam giác ABD có \(\widehat {{A^{}}}\) là góc tù.Nên BA < BD (trong tam giác tù, cạnh đối diện với góc tù là cạnh lớn nhất) (1)•Vì \(\widehat {B{\rm{D}}E}\) là góc ngoài của tam giác ADB tại đỉnh D \(\widehat {BDE} = \hat A + \widehat {ABD}\).Mà \(\widehat {{A^{}}}\) là góc tù.Do đó \(\widehat {B{\rm{D}}E}\) là góc tù.Xét tam giác EBD có \(\widehat {B{\rm{D}}E}\) là góc tù .Nên BD < BE (trong tam giác tù, cạnh đối diện với góc tù là cạnh lớn nhất) (2)•Vì \(\widehat {BEC}\) là góc ngoài của tam giác AEB tại đỉnh E nên \(\widehat {BEC} = \hat A + \widehat {ABE}\) Mà \(\widehat {{A^{}}}\)là góc tù.Do đó \(\widehat {BEC}\) là góc tù.Xét tam giác EBC có \(\widehat {BEC}\) là góc tù.Nên BE < BC (trong tam giác tù, cạnh đối diện với góc tù là cạnh lớn nhất) (3)Từ (1), (2) và (3) suy ra BA < BD < BE < BC.Vậy BA < BD < BE < BC.

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