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Giải bài 5 trang 87 sách bài tập toán 9 - Chân trời sáng tạo tập 2

Cho hình vuông ABCD cạnh a có O là giao điểm của hai đường chéo. Chứng minh có đường tròn (O; R) đi qua các đỉnh của hình vuông và có đường tròn (O; r) tiếp xúc với các cạnh của hình vuông. Tính theo a bán kính R và r.

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

Toán - Văn - Anh
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Đề bài

Cho hình vuông ABCD cạnh a có O là giao điểm của hai đường chéo. Chứng minh có đường tròn (O; R) đi qua các đỉnh của hình vuông và có đường tròn (O; r) tiếp xúc với các cạnh của hình vuông. Tính theo a bán ꦺkính R và r.

Phương pháp giải - Xem chi tiết

Đa giác đều có các cạnh bằng nhau và các góc bằng nhau.

Lời giải chi tiết

Hình vuông ABCD nội tiếp đường tròn (O; R) với R = OA = OB = OC = OD = \(\frac{{a\sqrt 2 }}{2}\).

Trong tam giác vuông cân AOD, vẽ đường cao OP, ta có r = OP = \(\frac{{AD}}{2} = \frac{a}{2}\).

Tương tự, ta có điểm O cách đều các cạnh của hình vuông một khoảng \(r = \frac{a}{2}\).

Do đó, đường tròn (O; r) với \(r = \frac{a}{2}\) tiếp xúc với các cạnh của hình vuông ABCD.

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