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Bài tập 15 trang 122 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác AMN vuông tại A. Tia phân giác của góc M và N gặp nhau tại I. Tia MI cắt AN tại R. Kẻ RT vuông góc với AI . Chứng minh rằng AT = RT.
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Đề bài

Cho tam giác AMN vuông tại A. Tia phân giác của góc M và N gặp nhau tại I. Tia MI cắt AN tại R. Kẻ RT vuông góc với AI (\(T \in AI\) ). Chứng minh rằng AT = RT.

Lời giải chi tiết

∆ABC có: I là giao điểm của hai đường phân giác của góc M và N (gt)Do đó: theo định lí về ba đường phân giác ta có I là điểm cách đều ba cạnh của tam giác ABC.=> AI là tia phân giác của \(\widehat {MAN}\)Do đó \(\widehat {RAI} = {{\widehat {MAN}} \over 2} = {{90^\circ } \over 2} = 45^\circ\)Ta có ∆ART vuông tại T có \(\widehat {RAT} = 45^\circ\)Do đó ∆ART vuông cân tại T => AT = RT.

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