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Giải bài 3.33 trang 83 SGK Toán 8 - Cùng khám phá

Cho hình vuông ABCD có cạnh dài
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Đề bài

Cho hình vuông ABCD có cạnh dài \(4\,cm\) và \(M,N,P,Q\) lần lượt là trung điểm các cạnh \(AB,BC,CD,AD.\) a) Chứng minh rằng tứ giác MNPQ là hình vuông. b) Tính diện tích hình vuông MNPQ.

Phương pháp giải - Xem chi tiết

Dựa vào tính chất và công thức tính diện tích hình vuông để chứng minh và tìm diện tích.

Lời giải chi tiết

a) Có hình vuông ABCD có cạnh dài \(4\,cm\) và \(M,N,P,Q\) lần lượt là trung điểm các cạnh \(AB,BC,CD,AD\)\( \Rightarrow AM = MB = BN = NC = DP = CP = AQ = DQ = 2cm.\)(1)Bốn tam giác vuông AMQ, BMQ, CNP, DPQ bằng nhau\( \Rightarrow QM = MN = NP = PQ\) (cạnh tương ứng bằng nhau).Có \(MP//AC;NQ//AB \Rightarrow MP \bot NQ\)(2)Từ (1) và (2) suy ra tứ giác MNPQ là hình vuông (dhnb).b) Xét tam giác vuông AMQ có\(Q{M^2} = A{M^2} + A{Q^2}\) (định lí Pythagore)\( \Rightarrow QM = \sqrt {{2^2} + {2^2}}  = 2\sqrt 2 cm\)Diện tích của hình vuông \(MNPQ\) là:\(2\sqrt 2 .2\sqrt 2  = 8c{m^2}\)Vậy diện tích hình vuông MNPQ bằng \(8\,c{m^2}.\)

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