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Giải bài tập 5.22 trang 103 SGK Toán 9 tập 1 - Kết nối tri thức

Cho góc xOy với đường phân giác Ot và điểm A trên cạnh Ox, điểm B trên cạnh Oy sao cho OA = OB. Đường thẳng qua A và vuông góc với Ox cắt Ot tại M. Chứng minh rằng OA và OB là hai tiếp tuyến cắt nhau của đường tròn (M; MA).

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

Cho góc xOy với đường phân giác Ot và điểm A trên cạnh Ox, điểm B trên cạnh Oy sao cho OA = OB. Đường thẳng qua A và vuông góc với Ox cắt Ot tại M. Chứng minh rằng OA và OB là hai tiếp tuyến cắt nhau của đường tròn (M; MA).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Ta có: OA là tiếp tuyến của đường tròn (M; MA) do OA \( \bot \) MA tại A. Xét cặp tam giác OAM và tam giác OBM, từ đó suy ra MA = MB và OB \( \bot \) MB. Hay OB là tiếp tuyến của đường tròn (M; MA).

Lời giải chi tiết

Ta có: OA là tiếp tuyến của đường tròn (M; MA) do OA \( \bot \) MA tại A. Xét tam giác OAM và tam giác OBM có: OM chung \(\widehat {{\rm{AOM}}} = \widehat {{\rm{BOM}}}\) (do OM là tia phân giác của góc \(\widehat {{\rm{AOB}}}\)) OA = OB Vậy \(\Delta {\rm{OAM}} = \Delta {\rm{OBM}}\) (c.g.c) Suy ra: MA = MB (hai cạnh tương ứng) \(\widehat {{\rm{OAM}}} = \widehat {{\rm{OBM}}} = 90^\circ \) (hai góc tương ứng) hay OB \( \bot \) MB Do đó OA là tiếp tuyến của đường tròn (M; MA) Vậy OA và OB là hai tiếp tuyến cắt nhau của (O).

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