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Câu 9 trang 13 SGK Hình học 11 Nâng cao

Cho góc nhọn
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Đề bài

Cho góc nhọn xOy và một điểm A nằm trong góc đó. Hãy xác định điểm B trên Ox và điểm C trên Oy sao cho tam giác ABC có chu vi nhỏ nhất

Lời giải chi tiết

Xét tam giác bất kì ABC có B và C lần lượt nằm trên hai tia Ox và Oy.Gọi A’ và A” là các điểm đối xứng với điểm A lần lượt qua các đường thẳng Ox và Oy.Ta có \(AB = A’B\) và \(AC = A”C\) ( do các \(△ABA’\) và \(△ACA”\) là các tam giác cân).Gọi \(2p\) là chu vi của tam giác ABC thì:\(2p = AB + BC + CA \)\(= A’B + BC + CA” ≥ A’A”\)Dấu “=” xảy ra khi bốn điểm \(A’, B, C, A”\) thẳng hàng.Suy ra để chu vi tam giác ABC bé nhất thì phải lấy B và C lần lượt là giao điểm của đoạn thẳng A’A” với hai tia Ox và Oy (các giao điểm đó tồn tại vì góc xOy nhọn)

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