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Bài tập 5 trang 130 Tài liệu dạy – học Toán 7 tập 2 - Hình học

Giải bài tập Cho tam giác ABC vuông tại A, kẻ đường cao AH. Trên tia đối của tia HA lấy điểm M sao cho H là trung điểm của AM.
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Đề bài

Cho tam giác ABC vuông tại A, kẻ đường cao AH. Trên tia đối của tia HA lấy điểm M sao cho H là trung điểm của AM. a) Chứng minh rằng tam giác ABM cân. b) Chứng minh tam giác ABC bằng tam giác MBC. c) Đường thẳng song song với AB kẻ từ M lần lượt cắt BC, AC tại I và N. Chứng minh đường thẳng AI vuông góc với MC.

Lời giải chi tiết

 

a) Ta có: \(BH \bot AM\) tại H (gt) và H là trung điểm của AM (gt)=> B thuộc đường trung trực của AM=> BA = BM=> ∆ABM cân tại Bb) ∆ABM cân tại B có BH ;à đường cao (\(BH \bot AM\) tại H)=> BH là đường phân giác của ∆ABM\( \Rightarrow \widehat {ABC} = \widehat {MBC}\)Xét ∆ABC và ∆MBC ta có:AB = BM (câu a)\(\widehat {ABC} = \widehat {MBC}\)BC (cạnh chung)Do đó: ∆ABC = ∆MBC (c.g.c).a) Ta có: MN // AB (gt)\(AB \bot AC\) (∆ABC vuông tại A) \( \Rightarrow MN \bot AC\)∆AMC có: CH là đường cao (\(CH \bot AM\) tại H)MN là đường cao (\(MN \bot AC\))CH cắt MN tại I (gt)Do đó I là trực tâm của ∆AMC => AI là đường cao của ∆AMCVậy \(AI \bot MC.\)

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