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Giải bài 8 trang 92 vở thực hành Toán 8 tập 2

Cho tam giác ABC cân tại A và M là trung điểm của cạnh BC. Lấy các điểm D, E lần lượt trên các cạnh AB, AC sao cho $\widehat{DME}=\widehat{ABC}$.

Tổng hợp đề t🐲hi học kì 2 lớp 8 tất cả các môn - Kế🥂t nối tri thức

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

Cho tam giác ABC cân tại A và M là trung điểm của cạnh BC. Lấy các điểm D, E lần lượt trên các cạnh AB, AC sao cho $\widehat{DME}=\widehat{ABC}$. a) Chứng minh $\Delta BDM\backsim \Delta CME$. b) Chứng minh DM là phân giác của góc BDE.

Phương pháp giải - Xem chi tiết

a) Chứng minh $\Delta BDM$ và $\Delta CME$ có hai cặp góc bằng nhau nên đồng dạng. b) Chứng minh $\Delta MDE\backsim \Delta CME$ suy ra cặp góc $\widehat{BDM}=\widehat{MDE}$.

Lời giải chi tiết

a) Xét hai tam giác BDM và CME, ta có:\(\widehat{DBM}=\widehat{ABC}=\widehat{BCA}=\widehat{MCE}\),$\widehat{BDM}={{180}^{0}}-\widehat{BMD}-\widehat{DBM}=\widehat{DMC}-\widehat{DME}=\widehat{EMC}$Vậy $\Delta BDM\backsim \Delta CME$ (g.g)b) Vì $\Delta BDM\backsim \Delta CME$ nên $\widehat{BDM}=\widehat{CME}$ và $\frac{DM}{ME}=\frac{BM}{CE}=\frac{MC}{CE}$, hay $\frac{DM}{MC}=\frac{ME}{CE}$.Xét hai tam giác MDE và CME, ta có:$\frac{DM}{MC}=\frac{ME}{CE}$ (theo chứng minh trên),$\widehat{DME}=\widehat{ABC}=\widehat{BCA}=\widehat{MCE}$.Vậy $\Delta MDE\backsim \Delta CME$ (c.g.c).Suy ra $\widehat{MDE}=\widehat{CME}=\widehat{BDM}$, hay DM là phân giác của góc BDE.

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