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Giải bài 1 trang 69 SGK Toán 8 – Cánh diều

Cho tam giác ABC có ba đường phân giácAD, BE, CF. Biết

ᩚᩚᩚᩚᩚᩚ⁤⁤⁤⁤ᩚ⁤⁤⁤⁤ᩚ⁤⁤⁤⁤ᩚ𒀱ᩚᩚᩚTổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Cho tam giác ABC có ba đường phân giác AD, BE, CF. Biết \(AB = 4,\,\,BC = 5,\,\,CA = 6\). Tính BD, CE, AF.

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Lời giải chi tiết

Có AD là đường phân giác trong tam giác ABC nên \(\frac{{AB}}{{AC}} = \frac{{DB}}{{DC}} = \frac{{DB}}{{DC}} = \frac{4}{6} = \frac{2}{3} \) nên \(DC = \frac{3}{2}DB\) Mà \(DB + DC = BC \) hay \(DB + \frac{3}{2}DB = 5 \) nên \(DB = 2\) Có BE là đường phân giác trong tam giác ABC nên \(\frac{{AE}}{{EC}} = \frac{{AB}}{{CB}} = \frac{4}{5} \) suy ra \(AE = \frac{4}{5}CE\) Mà \(AE + EC = AC \) hay \(\frac{4}{5}CE + CE = 6 \) nên \(CE = \frac{{10}}{3}\) Có CF là đường phân giác trong tam giác ABC nên \(\frac{{AF}}{{FB}} = \frac{{CA}}{{CB}} = \frac{6}{5} \) suy ra \(FB = \frac{6}{5}AF\) Mà \(AF + FB = AB \) hay \(AF + \frac{5}{6}AF = 4 \) nên \(AF = \frac{{24}}{{11}}\).

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