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Bài tập 34 trang 98 Tài liệu dạy – học Toán 7 tập 2

Giải bài tập Cho tam giác ABC vuông tại A có M là trung điểm của AC. Gọi E và F lần lượt là hình chiếu kẻ từ A và C đến các đường thẳng BM.
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Đề bài

Cho tam giác ABC vuông tại A có M là trung điểm của AC. Gọi E và F lần lượt là hình chiếu kẻ từ A và C đến các đường thẳng BM. Chứng minh: \(AB < {{BE + BF} \over 2} < BC\)

Lời giải chi tiết

 

Xét ∆AEM vuông tại E và ∆MCF vuông tại F ta có:AM = MC (M là trung điểm của AC)Và \(\widehat {EMA} = \widehat {CMF}\) (hai góc đối đỉnh)Do đó: ∆AEM = ∆CFM (cạnh huyền – góc nhọn)=> EM = MFBE + BF = BE + BM + MF = BE + BM + EM (vì MF = EM)= (BE + EM) + BM = BM + BM = 2BMDo đó \(BM = {{BE + BF} \over 2}(1)\)Mà \(BA \bot AC\) tại A,Mặt khác, ta có AM < AC => BM < BC (3)Từ (1), (2) và (3) ta có: \(AB < {{BE + BF} \over 2} < BC.\)

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