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Giải bài 13 trang 65 sách bài tập toán 8 - Chân trời sáng tạo tập 2

Cho tam giác ABC có ba góc nhọn. Trên cạnh AB lấy điểm D, trên cạnh AC lấy điểm E sao cho \(\widehat {ADE} = \widehat {ACB}\).

💙Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Chân trời sáng tạo

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

Cho tam giác ABC có ba góc nhọn. Trên cạnh AB lấy điểm D, trên cạnh AC lấy điểm E sao cho \(\widehat {ADE} = \widehat {ACB}\). a) Chứng minh rằng $\Delta AED\backsim \Delta ABC$. b) Tia phân giác của góc BAC cắt DE tại M và cắt BC tại N. Chứng minh rằng \(ME.NC = MD.NB\)

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về trường hợp đồng dạng thứ ba của hai tam giác (g.g) để chứng minh: Nếu hai góc của tam giác này lần lượt bằng hai góc của tam giác kia thì hai tam giác đó đồng dạng với nhau.

Lời giải chi tiết

a) Xét tam giác AED và tam giác ABC có:\(\widehat A\;chung,\widehat {ADE} = \widehat {ACB}\) (giả thiết)Do đó, $\Delta AED\backsim \Delta ABC\left( g.g \right)$b) Vì AM là tia phân giác của góc DAE nên \(\frac{{ME}}{{MD}} = \frac{{AE}}{{AD}}\)Vì AN là tia phân giác của góc BAC nên \(\frac{{NB}}{{NC}} = \frac{{AB}}{{AC}}\)Mà $\Delta AED\backsim \Delta ABC\left( cmt \right)$ nên \(\frac{{AE}}{{AB}} = \frac{{AD}}{{AC}}\) hay \(\frac{{AE}}{{AD}} = \frac{{AB}}{{AC}}\)Do đó, \(\frac{{ME}}{{MD}} = \frac{{NB}}{{NC}}\). Vậy \(ME.NC = MD.NB\)

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