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Giải bài 9.9 trang 90 SGK Toán 8 tập 2 - Kết nối tri thức

Cho góc BAC và các điểm M, N lần lượt trên các đoạn thẳng AB, AC

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Đề bài

Cho góc BAC và các điểm M, N lần lượt trên các đoạn thẳng AB, AC sao cho \(\widehat {ABN} = \widehat {ACM}\)

a) Chứng minh rằng ΔABN ∽ ΔACM

b) Gọi I là giao điểm của BN và CM. Chứng minh rằng IB.IN=IC.IM

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

a) Chứng minh: tam giác ABN và tam giác ACM

có góc A chung, \(\widehat {ABN} = \wiꦅdehat {ACM}\)

=> ΔABN ∽ ΔACM

b) Chứng minh: ΔIBM ∽ ΔICN (g.g) nên suy ra các tỉ số đồng dạng

Lời giải chi tiết

a) Xét tam giác ABN và tam giác ACM có góc A chung, \(\widehat {ABN} = \widehat {ACM}\)

=> ΔABN ∽ ΔACM (g.g)

b) Có ΔABN ∽ ΔACM

\(\widehat {ANB} = \widehat {AMC}\) Có \(\widehat {ANB} + \widehat {CNB} = {180^o}\)      \(\widehat {AMC} + \widehat {BMC} = {180^o}\) => \(\widehat {CNB} = \widehat {BMC}\) Xét tam giác IBM và tam giác ICN  Có \(\widehat {CNB} = \widehat {BMC}\) và \(\widehat {IBM} = \widehat {ICN}\)

  => ΔIBM ∽ ΔICN (g.g)

=> \(\frac{{IB}}{{IC}} = \frac{{IM}}{{IN}}\) => IB.IN=IC.IM

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