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Giải bài 12 trang 50 sách bài tập toán 8 - Chân trời sáng tạo tập 2

Cho tam giác ABC có \(I \in AB\) và \(K \in AC\). Kẻ IM//BK \(\left( {M \in AC} \right)\), KN//CI \(\left( {N \in AB} \right)\). Chứng minh MN//BC.

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

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Đề bài

Cho tam giác ABC có \(I \in AB\) và \(K \in AC\). Kẻ IM//BK \(\left( {M \in AC} \right)\), KN//CI \(\left( {N \in AB} \right)\). Chứng minh MN//BC. 

Phương pháp giải - Xem chi tiết

+ Sử dụng kiến thức về định lí Thalès trong tam giác để chứng minh: Nếu một đường thẳng song song với một cạnh của tam giác và cắt hai cạnh còn lại thì nó định ra trên hai cạnh đó các đoạn thẳng tương ứng tỉ lệ.  + Sử dụng kiến thức về định lí Thalès để chứng minh: Nếu một đường thẳng cắt hai cạnh của một tam giác và định ra trên hai cạnh này những đoạn thẳng tương ứng tỉ lệ thì đường thẳng đó song song với cạnh còn lại của tam giác.

Lời giải chi tiết

Tam giác ABK có IM//BK nên theo định lí Thalès ta có: \(\frac{{AI}}{{AB}} = \frac{{AM}}{{AK}}\)Tam giác AIC có KN//CI nên theo định lí Thalès ta có: \(\frac{{AN}}{{AI}} = \frac{{AK}}{{AC}}\)Do đó, \(\frac{{AI}}{{AB}}.\frac{{AN}}{{AI}} = \frac{{AM}}{{AK}}.\frac{{AK}}{{AC}} = \frac{{AM}}{{AC}}\), suy ra \(\frac{{AN}}{{AB}} = \frac{{AM}}{{AC}}\)Tam giác ABC có: \(\frac{{AN}}{{AB}} = \frac{{AM}}{{AC}}\) nên theo định lí Thalès đảo ta có MN//BC.

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