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Giải bài 7 trang 73 SGK Toán 10 tập 1 – Chân trời sáng tạo

Cho tam giác ABC có trọng tâm G và độ dài ba cạnh AB, BC, CA lần lượt là 15, 18, 27. a) Tính diện tích và bán kính đường tròn nội tiếp tam giác ABC. b) Tính diện tích tam giác GBC.

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Đề bài

Cho tam giác ABC có trọng tâm G và độ dài ba cạnh AB, BC, CA lần lượt là 15, 18, 27. a) Tính diện tích và bán kính đường tròn nội tiếp tam giác ABC. b) Tính diện tích tam giác GBC.

Phương pháp giải - Xem chi tiết

a) Tính r bằng công thức: \(S = p.r\). Trong đó S tính bởi công thức heron. b) Tìm a, từ đó suy ra R bằng định lí sin => Tính diện tích tam giác IBC

Lời giải chi tiết

a) Đặt \(a = BC,b = AC,c = AB.\)Ta có: \(p = \frac{1}{2}(15 + 18 + 27) = 30\)Áp dụng công thức heron, ta có:\({S_{ABC}} = \sqrt {30(30 - 15)(30 - 18)(30 - 27)}  = 90\sqrt 2 \)Và \(r = \frac{S}{p} = \frac{{90\sqrt 2 }}{{30}} = 3\sqrt 2 \)b) Gọi, H, K lần lượt là chân đường cao hạ từ A và G xuống BC, M là trung điểm BC.

G là trọng tâm tam giác ABC nên \(GM = \frac{1}{3}AM\)\(\begin{array}{l} \Rightarrow GK = \frac{1}{3}.AH\\ \Rightarrow {S_{GBC}} = \frac{1}{3}.\,{S_{ABC}} = \frac{1}{3}.90\sqrt 2  = 30\sqrt 2 .\end{array}\)

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