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Bài tập 38 trang 99 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC có M thuộc tia phân giác ngoài của góc C. Trên tia đối của tia CA lấy điểm I sao cho CI = CB.
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Đề bài

Cho tam giác ABC có M thuộc tia phân giác ngoài của góc C. Trên tia đối của tia CA lấy điểm I sao cho CI = CB. a) So sánh MI với MB. b) Chứng minh: MA + MB > AC + BC

Lời giải chi tiết

 

a) Gọi F là giao điểm của BC và MITa có \(\widehat {MCB} = \widehat {MCA} + \widehat {ACB}\) và \(\widehat {MCI} = \widehat {MCF} + \widehat {FCI}\)Mà \(\widehat {MCA} = \widehat {MCF}\)(CM là tia phân giác của \(\widehat {ACF}\))Và \(\widehat {ACB} = \widehat {FCI}\) (đối đỉnh). Do đó \(\widehat {MCB} = \widehat {MCI}\)Xét ∆MCB và ∆MCI ta có MC (cạnh chung)\(\widehat {MCB} = \widehat {MCI}\) và BC = CI (gt)Do đó ∆MCB = ∆MCI (c.g.c) => MB = MI.b) ∆AMI có MA + MI > AI (bất đằng thức trong tam giác) => MA + MI > AC + CIMà BC = CI, MB = MI (∆MCB = ∆MCI). Do đó MA + MB > AC + BC.

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