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Giải bài 9.15 trang 55 sách bài tập toán 8 - Kết nối tri thức với cuộc sống

Cho tam giác ABC và điểm O nằm trong tam giác. Lấy M, N, P là các điểm lần lượt trên các tia OA, OB, OC sao cho (OA = 3OM,OB = 3ON,OC = 3OP.)

💝Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Kết nối tri thức

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

Cho tam giác ABC và điểm O nằm trong tam giác. Lấy M, N, P là các điểm lần lượt trên các tia OA, OB, OC sao cho \(OA = 3OM,OB = 3ON,OC = 3OP.\) Chứng minh rằng $\Delta ABC\backsim \Delta MNP$ và tìm tỉ số đồng dạng

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về định lý (trường hợp đồng dạng cạnh – cạnh – cạnh) để chứng minh hai tam giác đồng dạng: Nếu ba cạnh của tam giác này tỉ lệ với ba cạnh của tam giác kia thì hai tam giác đó đồng dạng với nhau.

Lời giải chi tiết

Vì \(OA = 3OM,OB = 3ON,OC = 3OP\)  nên \(\frac{{OA}}{{OM}} = 3;\frac{{OB}}{{ON}} = 3;\frac{{OC}}{{OP}} = 3\) Tam giác OMN có: \(\frac{{OA}}{{OM}} = \frac{{OB}}{{ON}}\) nên AB//MN Do đó, \(\frac{{AB}}{{MN}} = \frac{{OA}}{{OM}} = 3\) Chứng minh tương tự ta có: \(\frac{{AC}}{{MP}} = 3;\frac{{BC}}{{NP}} = 3\) Tam giác ABC và tam giác MNP có: \(\frac{{AB}}{{MN}} = \frac{{AC}}{{MP}} = \frac{{BC}}{{NP}} = 3\) Do đó, $\Delta ABC\backsim \Delta MNP$ (c.c.c)  với tỉ số đồng dạng 3.

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