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Giải bài 11 trang 106 sách bài tập toán 9 - Cánh diều tập 1

Cho đường thẳng a và điểm O với khoảng cách từ điểm O đến đường thẳng a là 1 cm. Vẽ đường tròn tâm O bán kính 3 cm. a) Xác định vị trí tương đối của đường thẳng a và đường tròn (O). b) Gọi A và B là các giao điểm của đường thẳng a và đường tròn (O). Tính độ dài đoạn thẳng AB.

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

Toán - Văn - Anh
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Đề bài

Cho đường thẳng a và điểm O với khoảng cách từ điểm O đến đường thẳng a là 1 cm. Vẽ đường tròn tâm O bán kính 3 cm. a) Xác định vị trí tương đối của đường thẳng a và đường tròn (O). b) Gọi A và B là các giao điểm của đường thẳng a và đường tròn (O). Tính độ dài đoạn thẳng AB.

Phương pháp giải - Xem chi tiết

a) \(OH < R\): a và (O) cắt nhau

\(OH > R\): a và (O) không cắt nhau

\(OH = R\): a và (O) tiếp xúc nhau

b) Bước 1: Áp dụng định lý Pythagore trong tam giác vuông OAH để tính AH.

Bước 2: Chứng minh: \(AB = 2AH\)

Lời giải chi tiết

a) Kẻ \(OH \bot a\) tại H, khi đó ta có \(OH = 1\)cm, suy ra \(OH < R\) (vì \(R = 3\)cm). Vậy a và (O) cắt nhau. b) Xét tam giác BOA cân tại O (\(OB = OA = R\)) có đường cao OH (do \(OH \bot AB\)) đồng thời là đường trung tuyến nên \(AH = HB = \frac{{AB}}{2}\)  hay \(AB = 2AH\). Áp dụng định lý Pythagore trong tam giác vuông OAH ta có: \(AH = \sqrt {O{A^2} - O{H^2}}  = \sqrt {{3^2} - {1^2}}  = 2\sqrt 2 \)cm. Vậy \(AB = 2AH = 2.2\sqrt 2  = 4\sqrt 2 \)cm.

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

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