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Giải bài tập 3 trang 97 SGK Toán 9 tập 1 - Chân trời sáng tạo

Dây cung AB chia đường tròn (O) thành hai cung. Cung lớn có số đo bằng ba lần cung nhỏ. a) Tính số đo mỗi cung b) Chứng minh khoảng cách OH từ tâm O đến dây cung AB có độ dài bằng (frac{{AB}}{2}). Dây cung AB chia đường tròn (O) thành hai cung. Cung lớn có số đo bằng ba lần cung nhỏ. a) Tính số đo mỗi cung b) Chứng minh khoảng cách OH từ tâm O đến dây cung AB có độ dài bằng (frac{{AB}}{2}).

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

Toán - Văn - Anh
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Đề bài

Dây cung AB chia đường tròn (O) thành hai cung. Cung lớn có số đo bằng ba lần cung nhỏ. a) Tính số đo mỗi cung b) Chứng minh khoảng cách OH từ tâm O đến dây cung AB có độ dài bằng \(\frac{{AB}}{2}\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

- Đọc dữ kiện đề bài để vẽ hình. - Gọi \(\overset\frown{AnB}\) là cung nhỏ và \(\overset\frown{AmB}\) là cung lớn rồi lập biểu thức theo đề bài để tính. - Chứng minh H là trung điểm AB.

Lời giải chi tiết

a) Gọi \(\overset\frown{AnB}\) là cung nhỏ và \(\overset\frown{AmB}\) là cung lớn có sđ\(\overset\frown{AmB}\) = 3sđ\(\overset\frown{AnB}\) (gt)

Mà sđ\(\overset\frown{AmB}\) + sđ\(\overset\frown{AnB}\) = 360o

Do đó 4sđ\(\overset\frown{AnB}\) = 360o

sđ\(\overset\frown{AnB}\) = 360o: 4 = 90o

Vậy sđ\(\overset\frown{AmB}\) = 3sđ\(\overset\frown{AnB}\) = 3. 90o = 270o .

b) Ta có \(\widehat {AOB}\)= sđ\(\overset\frown{AnB}\) (góc ở tâm chắn cung AB)

suy ra \(\widehat {AOB}\)= 90o suy ra tam giác AOB vuông tại O.

Mà AO = OB = R nên tam giác AOB vuông cân tại O. Khi đó OH là đường cao đồng thời là đường trung tuyến. Tam giác AOB vuông tại O có OH là đường trung tuyến ứng với cạnh huyền nên OH = \(\frac{{AB}}{2}\).

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