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Giải bài 2 trang 48 sách bài tập toán 8 - Chân trời sáng tạo tập 2

Cho tam giác ABC có \(AB = 6cm,AC = 9cm,BC = 10cm\). Tia phân giác của góc BAC cắt BC tại D, tia phân giác của góc ngoài tại đỉnh A cắt BC tại E. Tính độ dài DB, DC, EB.

ꦍTổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Chân trời sáng tạo

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

Cho tam giác ABC có \(AB = 6cm,AC = 9cm,BC = 10cm\). Tia phân giác của góc BAC cắt BC tại D, tia phân giác của góc ngoài tại đỉnh A cắt BC tại E. Tính độ dài DB, DC, EB.

Phương pháp giải - Xem chi tiết

+ Sử dụng kiến thức về tính chất của đường phân giác của tam giác: Trong tam giác, đường phân giác của một góc chia cạnh đối diện thành hai đoạn thẳng tỉ lệ với hai cạnh kề đoạn ấy. + Định lí trên vẫn đúng đối với tia phân giác góc ngoài của tam giác.

Lời giải chi tiết

Vì AD là tia phân giác của góc BAC trong tam giác ABC nên theo tính chất của đường phân giác của tam giác ta có: \(\frac{{DB}}{{DC}} = \frac{{AB}}{{AC}} = \frac{6}{9} = \frac{2}{3}\) nên \(DB = \frac{2}{3}DC\)Ta có: \(BC = DB + DC = \frac{2}{3}DC + DC = \frac{5}{3}DC\)\(10 = \frac{5}{3}DC\) nên \(DC = 6cm\), suy ra: \(DB = 10 - 6 = 4\left( {cm} \right)\)Vì AE là tia phân giác của góc ngoài tại đỉnh A của tam giác ABC nên nên theo tính chất của đường phân giác của tam giác ta có: \(\frac{{EB}}{{EC}} = \frac{{AB}}{{AC}} = \frac{2}{3}\) nên \(EB = \frac{2}{3}EC\)Ta có: \(BC = EC - EB = EC - \frac{2}{3}EC = \frac{1}{3}EC\)\(10 = \frac{1}{3}EC\) nên \(EC = 30cm\), do đó \(EB = 20cm\)

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