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Giải bài tập 5.17 trang 114 SGK Toán 9 tập 1 - Cùng khám phá

Cho A là một điểm thuộc đường tròn (O), M là một điểm thuộc tiếp tuyến của (O) tại điểm A (M khác A). Đường tròn tâm M bán kính MA cắt (O) tại B (B khác A). Chứng minh rằng MB là một tiếp tuyến của (O).

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

Toán - Văn - Anh
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Đề bài

Cho A là một điểm thuộc đường tròn (O), M là một điểm thuộc tiếp tuyến của (O) tại điểm A (M khác A). Đường tròn tâm M bán kính MA cắt (O) tại B (B khác A). Chứng minh rằng MB là một tiếp tuyến của (O).

Phương pháp giải - Xem chi tiết

+ Chứng minh \(\widehat {MAO} = {90^o}\). + Chứng minh \(\Delta AMO = \Delta BMO\left( {c.c.c} \right)\) nên \(\widehat {MAO} = \widehat {MBO} = {90^o}\). + Suy ra \(MB \bot BO\) tại B. Mà B thuộc đường tròn (O) nên MB là tiếp tuyến của (O).

Lời giải chi tiết

Vì MA là tiếp tuyến của (O) nên \(MA \bot AO\) nên \(\widehat {MAO} = {90^o}\). Tam giác AMO và tam giác BMO có: \(OA = OB\) (bán kính (O)), \(MA = MB\) (bán kính (M)), OM chung. Do đó, \(\Delta AMO = \Delta BMO\left( {c.c.c} \right)\) nên \(\widehat {MAO} = \widehat {MBO} = {90^o}\) Suy ra \(MB \bot BO\) tại B. Mà B thuộc đường tròn (O) nên MB là tiếp tuyến của (O).

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