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Giải bài 10 trang 121 SGK Toán 8 tập 1 - Cánh diều

Cho hình vuông ABCD.

༺Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Cho hình vuông ABCD. Trên các cạnh AB, BC, CD, DA lần lượt lấy các điểm M, N, P, Q sao cho AM = BN = CP = DQ < AB. Chứng minh tứ giác MNPQ là hình vuông.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

ꦿChứng minh tứ giác MNPQ là hình thoi có một góc vuông nên MNPQ là hình vuông

Lời giải chi tiết

Ta có: AM = BN = CP = DQ (gt) AB = BC = CD = DA (ABCD là hình vuông) \(\Rightarrow\) BM = CN = DP = AQ \(\Rightarrow \Delta AMQ = \Delta BNM = \Delta CPN = \Delta DQP\)(hai cạnh góc vuông) Suy ra QM = MN = NP = PQ Suy ra MNPQ là hình thoi Do: \(\Delta AMQ = \Delta BNM \Rightarrow {\widehat M_1} = \widehat {BNM}\) (2 góc tương ứng) Mà: \(\widehat {BNM} + {\widehat M_3} = {90^0}\)(do \(\Delta BNM\)vuông tại B) \( \Rightarrow {\widehat M_1} + {\widehat M_3} = {90^0} \Rightarrow {\widehat M_2} = {180^0} - {\widehat M_1} - {\widehat M_3} = {180^0} - {90^0} = {90^0}\)

Vậy hình thoi MNPQ có một góc bằng 90o nên MNPQ là hình vuông

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