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Giải bài tập 2 trang 79 SGK Toán 9 tập 2 - Cánh diều

Cho đường tròn (I) nội tiếp tam giác ABC và lần lượt tiếp xúc với các cạnh BC, CA, AB tại M, N, P. Chứng minh (widehat {AIN} = widehat {PMN} = frac{1}{2}widehat {PIN.})

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

Toán - Văn - Anh
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Đề bài

Cho đường tròn (I) nội tiếp tam giác ABC và lần lượt tiếp xúc với các cạnh BC, CA, AB tại M, N, P. Chứng minh \(\widehat {AIN} = \widehat {PMN} = \frac{1}{2}\widehat {PIN.}\)

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Bước 1: Lý thuyết: Góc ở tâm bằng số đo cung bị chắn, góc nội tiếp bằng một nửa số đo cung bị chắn, suy ra \(\widehat {PMN} = \frac{1}{2}\widehat {PIN.}\) Bước 2: Sử dụng tính chất 2 tiếp tuyến cắt nhau để suy ra \(\widehat {AIN} = \frac{1}{2}\widehat {PIN.}\)

Lời giải chi tiết

Xét (I) có: \(\widehat {PIN}\) là góc ở tâm chắc cung NP nên \(\widehat {PIN}\)= sđ\(\overset\frown{NP}\). \(\widehat {PMN}\) là góc nội tiếp chắc cung NP nên \(\widehat {PMN}\) = \(\frac{1}{2}\)sđ\(\overset\frown{NP}\). Suy ra \(\widehat {PMN} = \frac{1}{2}\widehat {PIN.}\)(1) Ta lại có: \(IN \bot AC,IP \bot AB\) nên AB, AC là 2 tiếp tuyến của (I) nên IA là tia phân giác của góc PIN, hay \(\widehat {AIN} = \frac{1}{2}\widehat {PIN.}\)(2) Từ (1) và (2) ta có \(\widehat {AIN} = \widehat {PMN} = \frac{1}{2}\widehat {PIN.}\)

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Tham Gia Group Dành Cho Lớp 9 - Ôn Thi Vào Lớp 10 Miễn Phí

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