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Giải bài 12 trang 108 sách bài tập toán 9 - Cánh diều tập 2

Cho đa giác đều A1A2A3…An – 1 An (n > 3, n ∈ ℕ). Chứng minh các đường trung trực của các cạnh A1A2, A2A3, …, An – 1 An, AnA1 cùng đi qua một điểm.

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

Toán - Văn - Anh
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Đề bài

Cho đa giác đều A1A2A3…An – 1 An (n > 3, n ∈ ℕ). Chứng minh các đường trung trực của các cạnh A1A2, A2A3, …, An – 1 An, AnA1 cùng đi qua một điểm.

Phương pháp giải - Xem chi tiết

🌳Chứng minh tâm O của đa giác đều nằm trên đường trung trực. Sau đó chứng minh các đường trung trực cùng đi qua tâm của đa giác đều.

Lời giải chi tiết

Gọi O là tâm của đa giác đều A1A2A3…An – 1An.

Ta có OA1 = OA2 suy ra O nằm trên đường trung trực của cạnh A1A2.

Tương tự ta có O nằm trên các đường trung trực của các đoạn A2A3, …, An – 1An, AnA­1.

Suy ra các đường trung trực của các cạnh A1A2, A2A3, …, An – 1An, AnA­1ജ cùng đi qua một điểm, điểm đó là tâm của đa giác đều.

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