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Giải bài tập 5.44 trang 129 SGK Toán 9 tập 1 - Cùng khám phá

Cho MA và MB là hai tiếp tuyến của đường tròn (O; R) (A, B là hai tiếp điểm) sao cho \(\Delta \)MAB là tam giác đều. Khoảng cách OM bằng A. \(\frac{1}{2}R\). B. R. C. 2R. D. \(R\sqrt 2 \).

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

Toán - Văn - Anh
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Đề bài

Cho MA và MB là hai tiếp tuyến của đường tròn (O; R) (A, B là hai tiếp điểm) sao cho \(\Delta \)MAB là tam giác đều. Khoảng cách OM bằng A. \(\frac{1}{2}R\). B. R. C. 2R. D. \(R\sqrt 2 \).

Phương pháp giải - Xem chi tiết

+ Chứng minh \(\widehat {AMB} = {60^o}\). + Chứng minh MO là tia phân giác \(\widehat {AMB}\), nên \(\widehat {AMO} = \frac{1}{2}\widehat {AMB}\). + Chứng minh tam giác AOM vuông tại M nên \(AO = MO.\sin AMO\), từ đó tính được MO.

Lời giải chi tiết

Vì tam giác MAB đều nên \(\widehat {AMB} = {60^o}\). Vì MA và MB là tiếp tuyến của (O) nên MO là tia phân giác \(\widehat {AMB}\), nên \(\widehat {AMO} = \frac{1}{2}\widehat {AMB} = \frac{1}{2}{.60^o} = {30^o}\) Vì MA là tiếp tuyến của (O) nên \(MA \bot AO\). Do đó, tam giác MAO vuông tại A. Suy ra, \(AO = MO.\sin AMO\) nên \(MO = \frac{{AO}}{{\sin AMO}} = \frac{R}{{\sin {{30}^o}}} = 2R\). Chọn C

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