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Bài 2 trang 171 Tài liệu dạy – học Toán 7 tập 1

Giải bài tập Cho tam giác ABC vuông tại A có AB = 9 cm, AC = 12 cm.
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Đề bài

Cho tam giác ABC vuông tại A có AB = 9 cm, AC = 12 cm. a) Tính độ dài BC. b) Gọi M là trung điểm của BC, trên tia đối của tia MA lấy điểm D sao cho M là trung điểm của AD. Chứng minh rằng \(\Delta AMB = \Delta DMC.\) c) Chứng minh rằng tam giác ACD vuông.

Lời giải chi tiết

 

a)Tam giác ABC vuông tại A (gt) \(\Rightarrow B{C^2} = A{B^2} + A{C^2}\)   (định lý Pythapore)Do đó: \(B{C^2} = {9^2} + {12^2} = 81 + 144 = 225.\)Mà BC > 0 nên \(BC = \sqrt {225}  = 15(cm).\)b) Xét tam giác AMB và DMC ta có:AM = DM (giả thiết)BM = CM (M là trung điểm của BC)\(\widehat {AMB} = \widehat {CMD}\)   (hai góc đối đỉnh)Do đó: \(\Delta AMB = \Delta DMC(c.g.c)\)c) Ta có: \(\widehat {MBA} = \widehat {MCD}(\Delta AMB = \Delta DMC)\)Mà hai góc MBA và MCD so le trong. Do đó: AB // CD.Mà \(AB \bot AC(gt) \Rightarrow AC \bot CD.\)   Vậy tam giác ACD vuông tại C.

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