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Giải bài 6 trang 75 SBT toán 10 - Cánh diều

Cho tam giác ABC có \(\widehat B = {60^0},\widehat C = {105^0}\) và \(BC = 15\). Tính độ dài cạnh AC và bán kính R của đường tròn ngoại tiếp tam giác ABC (làm tròn kết quả đến hàng phần mười)

Tổng hợp đề thi học kì 2 lớp 1𓃲0 tất cꩵả các môn - Cánh diều

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Đề bài

Cho tam giác ABC có \(\widehat B = {60^0},\widehat C = {105^0}\) và \(BC = 15\). Tính độ dài cạnh AC và bán kính R của đường tròn ngoại tiếp tam giác ABC (làm tròn kết quả đến hàng phần mười)

Phương pháp giải - Xem chi tiết

Tính góc A và sử dụng định lí sin để tính độ dài AC và bán kính R

Lời giải chi tiết

Ta có: \(\widehat A = {180^0} - (\widehat B + \widehat C) = {15^0}\)

Áp dụng định lí sin cho ∆ABC ta có: \(\left\{ \begin{array}{l}\frac{{AC}}{{\sin B}} = \frac{{BC}}{{\sin {\rm{A}}}} \Rightarrow AC = \frac{{BC.\sin B}}{{\sin {\rm{A}}}} = \frac{{15.\sin {{60}^0}}}{{\sin {{15}^0}}} \approx 50,2\\\frac{{BC}}{{\sin {\rm{A}}}} = 2R \Rightarrow R = \frac{{BC}}{{2\sin {\rm{A}}}} = \frac{{15}}{{2\sin {🦩{15}^0}}} \approx 29\end{array} \right.\)

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