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Giải bài tập 5.2 trang 86 SGK Toán 9 tập 1 - Kết nối tri thức

Cho tam giác ABC vuông tại A có AB = 3 cm, AC = 4 cm. Chứng minh rằng các điểm A, B, C thuộc cùng một đường tròn. Tính bán kính của đường tròn đó.

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

Toán - Văn - Anh
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Đề bài

Cho tam giác ABC vuông tại A có AB = 3 cm, AC = 4 cm. Chứng minh rằng các điểm A, B, C thuộc cùng một đường tròn. Tính bán kính của đường tròn đó.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

- Từ định lý Pythagore tính được độ dài cạnh BC. - Từ tính chất đường trung tuyến trong tam giác vuông suy ra A, B, C thuộc đường tròn đường kính BC.

Lời giải chi tiết

Áp dụng định lý Pythagore ta có: \(B{C^2} = A{B^2} + A{C^2} = {3^2} + {4^2} = 25 \Rightarrow BC = 5\)(cm) Gọi O là trung điểm của cạnh BC. Ta có AO là trung tuyến ứng với cạnh huyền nên \(OA = \frac{{BC}}{2} = OB = OC = 2,5\)(cm). Suy ra A, B, C cùng thuộc đường tròn bán kính OB = 2,5 cm.

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