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Giải bài 6 trang 73 SGK Toán 10 tập 1 – Chân trời sáng tạo

Cho tam giác ABC có AB = 6,AC = 8 và A =60 a) Tính diện tích tam giác ABC. b) Gọi I là tâm đường tròn ngoại tiếp tam giác ABC. Tính diện tích tam giác IBC.

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Đề bài

Cho tam giác ABC có \(AB = 6,AC = 8\) và \(\widehat A = {60^o}.\) a) Tính diện tích tam giác ABC. b) Gọi I là tâm đường tròn ngoại tiếp tam giác ABC. Tính diện tích tam giác IBC.

Phương pháp giải - Xem chi tiết

a) Tính diện tích bằng công thức: \(S = \frac{1}{2}bc\sin A\) b) Tìm a, từ đó suy ra R bằng định lí sin => Tính diện tích tam giác IBC

Lời giải chi tiết

Đặt \(a = BC,b = AC,c = AB.\)a) Áp dụng công thức \(S = \frac{1}{2}bc\sin A\), ta có: \({S_{ABC}} = \frac{1}{2}.8.6.\sin {60^o} = \frac{1}{2}.8.6.\frac{{\sqrt 3 }}{2} = 12\sqrt 3 \)b) Áp dụng định lí cosin cho tam giác ABC ta được:\(\begin{array}{l}B{C^2} = {a^2} = {8^2} + {6^2} - 2.8.6.\cos {60^o} = 52\\ \Rightarrow BC = 2\sqrt {13} \end{array}\)Xét tam giác IBC ta có:Góc \(\widehat {BIC} = 2.\widehat {BAC} = {120^o}\)(góc ở tâm và góc nội tiếp cùng chắn một cung)\(IB = IC = R = \frac{a}{{2\sin A}} = \frac{{2\sqrt {13} }}{{2.\frac{{\sqrt 3 }}{2}}} = \frac{{2\sqrt {39} }}{3}.\)\( \Rightarrow {S_{IBC}} = \frac{1}{2}.\frac{{2\sqrt {39} }}{3}.\frac{{2\sqrt {39} }}{3}\sin {120^o} = \frac{{13\sqrt 3 }}{3}.\)

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