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Giải bài 1 trang 48 sách bài tập toán 8 - Chân trời sáng tạo tập 2

Cho tam giác ABC vuông tại A. Tia phân giác của góc BAC cắt BC tại D. Cho biết \(DB = 15cm,DC = 20cm\). Tính độ dài AB, AC.

ဣTổng 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 vuông tại A. Tia phân giác của góc BAC cắt BC tại D. Cho biết \(DB = 15cm,DC = 20cm\). Tính độ dài AB, AC.

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.

Lời giải chi tiết

Ta có: \(BC = DC + DB = 35\left( {cm} \right)\)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: \(\frac{{AB}}{{AC}} = \frac{{BD}}{{DC}} = \frac{{15}}{{20}} = \frac{3}{4}\), suy ra: \(AB = \frac{3}{4}AC\)Áp dụng định lí Pythagore vào tam giác ABC vuông tại A ta có: \(B{C^2} = A{B^2} + A{C^2}\), \({35^2} = \frac{9}{{16}}A{C^2} + A{C^2}\),\(A{C^2} = 784\) nên \(AC = 28cm\), do đó \(AB = \frac{3}{4}.28 = 21\left( {cm} \right)\)

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