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Giải bài 26 trang 99 sách bài tập toán 8 - Cánh diều

Cho hình thoi (ABCD) có góc (B) tù. Kẻ (BE) vuông góc (AD) tại (E), (BF) vuông góc với (CD) tại (F).

𝔉Tổng 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 thoi \(ABCD\) có góc \(B\) tù. Kẻ \(BE\) vuông góc \(AD\) tại \(E\), \(BF\) vuông góc với \(CD\) tại \(F\). Gọi \(M,N\) lần lượt là giao điểm của \(BE,BF\) với \(AC\). Chứng minh tứ giác \(BMDN\) là hình thoi.

Phương pháp giải - Xem chi tiết

Dựa vào tính chất của hình thoi: Trong một hình thoi: -         Các cạnh đối song song -         Các góc đối bằng nhau -         Hai đường chéo vuông góc với nhau và cắt nhau tại trung điểm của mỗi đường -         Hai đường chéo là các đường phân giác của các góc ở đỉnh.

Lời giải chi tiết

Gọi \(O\) là giao điểm của \(AC\) và \(BD\) Do \(ABCD\) là hình thoi nên \(AC\) vuông góc với \(BD\) tại trung điểm \(O\) của \(BD\). Suy ra \(AC\) là đường trung trực của \(BD\). Do đó \(BM = DM,BN = DN\). Do \(ABCD\) là hình thoi nên \(BA = BC,\widehat {BAE} = \widehat {BCF}\). Suy ra \(\Delta ABE = \Delta BCF\) (cạnh huyền – góc nhọn kề) Do đó \(\widehat {ABE} = \widehat {CBF}\). Mà \(\widehat {ABD} = \widehat {CBD}\), suy ra \(\widehat {MBO} = \widehat {NBO}\). \(\Delta MBO = \Delta NBO\) (cạnh góc vuông – góc nhọn). suy ra \(BM = BN\) Mà \(BM = DM\) và \(BN = DN\), suy ra \(BM = DM = BN = DN\). Tứ giác \(BMDN\) có \(BM = DM = BN = DN\) nên \(BMDN\) là hình thoi.

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