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Giải Bài 57 trang 86 sách bài tập toán 7 - Cánh diều

Cho tam giác ABC vuông tại A, tia phân giác của góc B cắt AC ở D. So sánh độ dài AD và DC.

𒉰Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Cánh diều

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Đề bài

Cho tam giác ABC vuông tại A, tia phân giác của góc B cắt AC ở D. So sánh độ dài AD và DC.

Phương pháp giải - Xem chi tiết

Chứng minh: AD = HD và HD < DC suy ra AD < DC

Lời giải chi tiết

 

Kẻ DH ⊥ BC.Vì BD là tia phân giác của góc ABC nên \({\hat B_1} = {\hat B_2}\)Xét ∆DAB và ∆DHB có:\(\widehat {BAD} = \widehat {BHD}\left( { = 90^\circ } \right)\)BD là cạnh chung,\({\hat B_1} = {\hat B_2}\) (chứng minh trên)Do đó ∆DAB = ∆DHB (cạnh huyền – góc nhọn).Suy ra AD = HD (hai cạnh tương ứng) (1)Vì ∆DHC vuông tại H nên HD < DC (trong tam giác vuông, cạnh huyển là cạnh lớn nhất) (2)Từ (1) và (2) suy ra AD < DC.Vậy AD < DC.

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