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Giải bài 2 trang 89 sách bài tập toán 9 - Chân trời sáng tạo tập 1

Từ điểm P ở ngoài đường tròn (O; R) vẽ hai tiếp tuyến tiếp xúc với (O) tại A và B. Đoạn thẳng OP cắt (O) tại Q (Hình 10). Cho biết PB = 8, PQ = 4. Tính R và số đo (widehat {AOB}).

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

Toán - Văn - Anh
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Đề bài

Từ điểm P ở ngoài đường tròn (O; R) vẽ hai tiếp tuyến tiếp xúc với (O) tại A và B. Đoạn thẳng OP cắt (O) tại Q (Hình 10). Cho biết PB = 8, PQ = 4. Tính R và số đo \(\widehat {AOB}\).

Phương pháp giải - Xem chi tiết

Áp dụng định lí Pythagore vào tam giác OPB để tính R. Sử dụng tỉ số lượng giác, từ đó tính góc BOP. Khi đó ta tính được góc AOB.

Lời giải chi tiết

Trong \(\Delta OPB\) vuông tại B, ta có OP2 = OB2 + PB2, suy ra (R + 4)2 = R2 + 82, suy ra R = 6.

OP2 = OB2 + PB2 suy ra (R + 4)2 = R2 + 82, suy ra R = 6.

\(\sin \widehat {BOP} = \frac{{PB}}{{OP}} = \frac{8}{{6 + 4}} = \frac{4}{5}\) suy ra \(\widehat {BOP} \approx {53^o}\). Ta lại có \(\widehat {AOB} = 2\widehat {BOP} \approx {2.53^o} = {106^o}.\)

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

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