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Bài tập 3 trang 127 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC nhọn (AB < AC), vẽ đường cao AH. Đường trung trực của cạnh BC cắt AC tại M, cắt BC tại N.
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Đề bài

Cho tam giác ABC nhọn (AB < AC), vẽ đường cao AH. Đường trung trực của cạnh BC cắt AC tại M, cắt BC tại N. a) Chứng minh rằng \(\widehat {BMN} = \widehat {HAC}\) b) Kẻ \(MI \bot AH\left( {I \in AH} \right)\) , gọi K là giao điểm của AH với BM. Chứng minh rằng I là trung điểm của AK.

Lời giải chi tiết

 

a) Ta có MN là đường trung trực của BC (gt) \( \Rightarrow MN \bot BC.\)Mà \(AH \bot BC\) (AH là đường cao của tam giác ABC). Nên MN // AH.M thuộc đường trung trực của BC (gt).=> MB = MC => ∆MBC cân tại MDo đó MN là đường phân giác của ∆MBC\( \Rightarrow \widehat {BMN} = \widehat {NMC}.\)Mà \(\widehat {NMC} = \widehat {HAC}\) (hai góc đồng vị và MN // AH)Vậy \(\widehat {BMN} = \widehat {HAC}.\)b) Ta có \(\widehat {BMN} = \widehat {HAC} \Rightarrow \widehat {BMN} = \widehat {KAM}\)Mà \(\widehat {BMN} = \widehat {AKM}\) (hai góc so le trong và MN // AH). Nên \(\widehat {KAM} = \widehat {AKM}.\)Do đó ∆AKM cân tại M.Lại có MI là đường cao của tam giác AKM (\(MI \bot AK\) tại I).Nên MI cũng là đường trung tuyến của tam giác AKM.Vậy I là trung điểm của AK.

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