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Giải bài 4 trang 69 SGK Toán 8 – Cánh diều

Cho hình thoi ABCD (Hình 4). Điểm M thuộc cạnh AB thỏa mãn

🍰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 thoi ABCD (Hình 4💟). Điểm M thuộc cạnh AB thỏa mãn \(AB = 3AM\). Hai đoạn thẳng AC và DM cắt nhau tại N. Chứng minh \(ND = 3MN\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Sử dụng tính chất đường phân giác để chứng minh yêu cầu bài toán.

Lời giải chi tiết

Gọi giao điểm hai đường chéo của hình thoi là O. Khi đó AC vuông góc với BD tại O. Vì ABCD là hình thoi nên \(AB = AD\) hay tam giác ABD cân tại A. Khi đó AO vừa là đường cao, vừa là phân giác của tam giác ABD. Xét tam giác AMD với AN là đường phân giác, ta có: \(\frac{{ND}}{{NM}} = \frac{{AD}}{{AM}}\,\,\left( 1 \right)\) (Tính chất đường phân giác) Mà \(AB = 3AM \Rightarrow \frac{{AB}}{{AM}} = 3 \Rightarrow \frac{{AD}}{{AM}} = 3\,\,\left( 2 \right)\) Từ (1) và (2) ta có: \(\frac{{ND}}{{NM}} = 3 \Rightarrow ND = 3NM\).

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