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Bài tập 16 trang 97 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Chứng minh rằng trong một tam giác cân, độ dài đoạn thẳng nối đỉnh với một điểm bất kì của cạnh đáy thì luôn luôn nhỏ hơn hoặc bằng độ dài cạnh bên.
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Đề bài

Chứng minh rằng trong một tam giác cân, độ dài đoạn thẳng nối đỉnh với một điểm bất kì của cạnh đáy thì luôn luôn nhỏ hơn hoặc bằng độ dài cạnh bên.

Lời giải chi tiết

 

Xét ∆ABC cân tại A, D là điểm trên cạnh BC.Cần chứng minh rằng AD ≤ AB* Nếu \(D \equiv B\) thì AD = AB* Nếu \(D \equiv C\) thì AD = AC = AB* Nếu D khác B và CVẽ \(AH \bot BC\) tại H* Nếu D thuộc đoạn thẳng HB (D khác B)\(D \equiv H\) thì AD = AH < AB (đường vuông góc ngắn hơn mọi đường xiên).\(D \ne H\) thì HD < HB, HD và HB lần lượt là hình chiếu của AD và AB trên BCDo đó AD < AC.* Nếu D thuộc đoạn thẳng HCTương tự trên có AD < AC = ABTóm lại AD ≤ AB.

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