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Giải bài tập 9.8 trang 76 SGK Toán 9 tập 2 - Kết nối tri thức

Cho tam giác đều ABC nội tiếp đường tròn (O). Biết rằng đường tròn (O) có bán kính bằng 3cm. Tính diện tích tam giác ABC.

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

Cho tam giác đều ABC nội tiếp đường tròn (O). Biết rằng đường tròn (O) có bán kính bằng 3cm. Tính diện tích tam giác ABC.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

+ Vì tam giác ABC đều nội tiếp đường tròn (O) nên O là trọng tâm, trực tâm của tam giác ABC. + Gọi H là giao điểm của AO và BC nên AH là trung trực đồng thời là đường cao trong tam giác đều ABC. Do đó: \(OA = \frac{{BC\sqrt 3 }}{3}\), từ đó tính được BC. + Diện tích tam giác ABC: \(S = \frac{1}{2}AH.BC\).

Lời giải chi tiết

Vì tam giác ABC đều nội tiếp đường tròn (O) nên O là trọng tâm, trực tâm của tam giác ABC. Gọi H là giao điểm của AO và BC nên AH là trung trực đồng thời là đường cao, đường trung tuyến trong tam giác đều ABC. Do đó: \(OA = \frac{{BC\sqrt 3 }}{3} \Rightarrow BC = \sqrt 3 OA = 3\sqrt 3 \left( {cm} \right)\) Vì O là trọng tâm của tam giác ABC, AH là đường trung tuyến của tam giác ABC nên \(AH = \frac{3}{2}OA = \frac{3}{2}.3 = \frac{9}{2}\left( {cm} \right)\) Diện tích tam giác ABC là: \(S = \frac{1}{2}AH.BC = \frac{1}{2}.\frac{9}{2}.3\sqrt 3  = \frac{{27\sqrt 3 }}{4}\left( {c{m^2}} \right)\)

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Tham Gia Group Dành Cho Lớp 9 - Ôn Thi Vào Lớp 10 Miễn Phí

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