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Bài tập 2 trang 127 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Hai làng A và B nằm cùng phía bên bờ sông (hình 77). Hằng ngày, các em học sinh phải vượt sông đến trường ở bên kia sông trên những chiếc bè gỗ. Để đảm bảo an toàn cho học sinh, người ta dự định xây một cây cầu bắc ngang sông. Hãy tìm địa điểm C trên bờ sông để xây cầu sao cho tổng quãng đường từ đầu cầu đến hai làng A và B là ngắn nhất.
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Đề bài

Hai làng A và B nằm cùng phía bên bờ sông (hình 77). Hằng ngày, các em học sinh phải vượt sông đến trường ở bên kia sông trên những chiếc bè gỗ. Để đảm bảo an toàn cho học sinh, người ta dự định xây một cây cầu bắc ngang sông. Hãy tìm địa điểm C trên bờ sông để xây cầu sao cho tổng quãng đường từ đầu cầu đến hai làng A và B là ngắn nhất.

 

Lời giải chi tiết

 

Gọi bờ sông là đường thẳng d, D là giao điểm sao cho d là đường trung trực của đoạn thẳng AD, E là giao điểm của BD và d.Nếu \(C \equiv E\) ta có CA + CB = BD.Nếu C ≠ E ta có CA = CD.(vì C thuộc đường trung trực của AD)Xét ∆BCD ta có: CD + CB > BD.Do đó CA + CB > BD.Vậy khi C là giao điểm của d và BD thì CA+CB ngắn nhất.

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