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Giải bài 9.11 trang 52 sách bài tập toán 8 - Kết nối tri thức với cuộc sống

Cho tam giác ABC với \(AB = 6cm,AC = 9cm.\) a) Lấy các điểm M, N lần lượt trên AB, AC sao cho \(AM = 4cm,AN = 6cm\).

ꦑTổng 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 với \(AB = 6cm,AC = 9cm.\) a) Lấy các điểm M, N lần lượt trên AB, AC sao cho \(AM = 4cm,AN = 6cm\). Chứng minh rằng $\Delta AMN\backsim \Delta ABC$ và tìm tỉ số đồng dạng. b) Lấy điểm P trên cạnh AC sao cho \(AP = 4cm.\) Chứng minh rằng $\Delta APB\backsim \Delta ABC$

Phương pháp giải - Xem chi tiết

+ Sử dụng kiến thức định lí (một trường hợp đặc biệt của hai tam giác đồng dạng) để chứng minh hai tam giác đồng dạng để chứng minh $\Delta AMN\backsim \Delta ABC$: Nếu một đường thẳng cắt hai cạnh của một tam giác và song song với cạnh còn lại thì nó tạo thành một tam giác mới đồng dạng với tam giác đã cho. + Sử dụng kiến thức trường hợp bằng nhau của tam giác (cạnh – góc – cạnh) để chứng minh \(\Delta APB = \Delta AMN\)

Lời giải chi tiết

a) Xét tam giác ABC có: \(\frac{{AM}}{{AB}} = \frac{{AN}}{{AC}}\left( {do\;\frac{4}{6} = \frac{6}{9}} \right)\)  nên MN//BC (định lí Thalès đảo)Do đó, $\Delta AMN\backsim \Delta ABC$ với tỉ số đồng dạng \(\frac{2}{3}\) (1)b) Tam giác APB và tam giác AMN có:\(AP = AM\left( { = 4cm} \right),\widehat A\;chung,AB = AN\;\left( { = 6cm} \right)\)Do đó, \(\Delta APB = \Delta AMN\left( {c.g.c} \right)\) (2)Từ (1) và (2) ta có: $\Delta APB\backsim \Delta ABC$

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