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Giải bài 6.11 trang 47 SGK Toán 8 - Cùng khám phá

Tam giác \(ABC\) vuông tại \(B\) có độ dài hai cạnh góc vuông là
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Đề bài

Tam giác \(ABC\) vuông tại \(B\) có độ dài hai cạnh góc vuông là \(AB = 9cm,BC = 12cm.\) Tính độ dài \(BD\) với \(D\) là chân đường phân giác kẻ từ đỉnh \(A\) của tam giác \(ABC.\) 

Phương pháp giải - Xem chi tiết

Dựa vào tính chất đường phân giác của một 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 hai đoạn ấy. và định lí Pythagore để tính độ dài BD.

Lời giải chi tiết

Áp dụng định lí Pythagore trong tam giác vuông ta có: \(\begin{array}{l}A{C^2} = A{B^2} + B{C^2} = {9^2} + {12^2} = 225\\ = > AC = 15\end{array}\)Áp dụng tính chất đường phân giác trong tam giác ta có: \(\frac{{BD}}{{DC}} = \frac{{AB}}{{AC}} \Leftrightarrow \frac{x}{{12 - x}} = \frac{9}{{15}} \Leftrightarrow 15x = 9\left( {12 - x} \right) \Leftrightarrow 15x = 108 - 9x \Rightarrow x = 4,5\)Vậy \(BD = 4,5;DC = 12 - 4,5 = 7,5\) 

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