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Bài tập 28 trang 123 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC vuông cân tại A. Lấy điểm E thuộc cạnh AC. Trên tia đối của tia AB lấy điểm D sao cho AD = AE. Chứng minh:
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Đề bài

Cho tam giác ABC vuông cân tại A. Lấy điểm E thuộc cạnh AC. Trên tia đối của tia AB lấy điểm D sao cho AD = AE. Chứng minh: a) DE vuông góc với BC. b) BE vuông góc với DC.

Lời giải chi tiết

 

a) Gọi H là giao điểm của DE và BC.Ta có: \(\widehat {ADE} + \widehat {AED} = 90^\circ\) (∆ADE vuông tại A)\(\widehat {ADE} = \widehat {ECH}( = 45^\circ )\)\(\widehat {AED} = \widehat {HEC}\) (hai góc đối đỉnh)Do đó: \(\widehat {ECH} + \widehat {HEC} = 90^\circ\)Mà \(\widehat {ECH} + \widehat {HEC} + \widehat {EHC} = 180^\circ\) (tổng ba góc trong một tam giác)Nên \(90^\circ  + \widehat {EHC} = 180^\circ  \Rightarrow \widehat {EHC} = 90^\circ  \Rightarrow EH \bot BC \Rightarrow DE \bot BC.\)b) ∆BDC có: DE là đường cao \((DE \bot BC),\)CA là đường cao \((CA \bot AB,D \in BA)\) và DE cắt CA tại E (gt)Do đó E là trực tâm của ∆BDC.Vậy BE là đường cao của tam giác ABC. Nên \(BE \bot DC.\)

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