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Giải bài 15 trang 79 SBT toán 10 - Cánh diều

Cho tam giác ABC có \(\widehat B = {60^0},BC = 8,AB + AC = 12\). Tính độ dài các cạnh AB, AC

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Đề bài

Cho tam giác ABC có \(\widehat B = {60^0},BC = 8,AB + AC = 12\). Tính độ dài các cạnh AB, AC           

Phương pháp giải - Xem chi tiết

Bước 1: Biểu diễn AB hoặc AC theo cạnh còn lại

Bước 2: Áp dụng định lí cosin cho ∆ABC, lập PT với ẩn AB hoặc AC

Bước 3: Giải PT ở bước 2 để tìm độ dài cạnh AB, AC rồi kết luận

Lời giải chi tiết

Theo giả thiết, \(AB + AC = 12 \Rightarrow AC = 12 - AB\)

Áp dụng định lí cosin cho ∆ABC ta có:

\(\begin{array}{l}A{C^2} = A{B^2} + B{C^2} - 2.AB.BC.\cos B\\ \Leftrightarrow {(12 - AB)^2} = A{B^2} + B{C^2} - 2.AB.BC.\cos B\\ \Leftrightarrow 144 - 24.AB + A{B^2} = A{B^2} + B{C^2} - 2.AB.BC.\cos B\end{array}\)\( \Leftrightarrow B{C^2} - 2.AB.BC.\cos B + 24AB - 144 = 0\)\( \Leftrightarrow {8^2} - 2.AB.8.\cos {60^0} + 24AB - 144 = 0\)\( \Leftrightarrow 16AB - 80 = 0 \Leftrightarrow AB = 5\)\( \Rightarrow \) \(AC = 12 - AB = 7\)

Vậy AB = 5, AC = 7

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