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Giải bài 9 trang 89 vở thực hành Toán 8 tập 2

Cho tam giác ABC và hai điểm P, Q lần lượt nằm trên các cạnh AB và AC sao cho \(\widehat{ABQ}=\widehat{ACP}\).

🐠Tổng 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 và hai điểm P, Q lần lượt nằm trên các cạnh AB và AC sao cho \(\widehat{ABQ}=\widehat{ACP}\). Chứng minh rằng $\Delta APC\backsim \Delta AQB$ và $\Delta APQ\backsim ACB$.

Phương pháp giải - Xem chi tiết

Chứng minh dựa vào các trường hợp đồng dạng của hai tam giác.

Lời giải chi tiết

Xét hai tam giác APC và AQB, ta có: $\widehat{ACP}=\widehat{ABQ}$ (theo giả thiết), $\widehat{PAC}=\widehat{QAB}$ (góc chung).Do đó $\Delta APC\backsim AQB$ (g.g).Vì $\Delta APC\backsim AQB$ nên $\frac{AP}{AQ}=\frac{AC}{AB}$, hay $\frac{AP}{AC}=\frac{AQ}{AB}$.Xét hai tam giác APQ và ACB, ta có:$\frac{AP}{AC}=\frac{AQ}{AB}$ (theo chứng minh trên), $\widehat{PAQ}=\widehat{CAB}$ (góc chung).Do đó $\Delta APQ\backsim \Delta ACB$ (c.g.c).

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