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Giải bài tập 2 trang 97 SGK Toán 9 tập 1 - Chân trời sáng tạo

Cho tam giác đều ABC. Vẽ nửa đường tròn đường kính BC cắt cạnh AB và AC lần lượt tại D và E. Hãy so sánh các cung (oversetfrown{BD};oversetfrown{BE};oversetfrown{EC}).

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

Toán - Văn - Anh
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Đề bài

Cho tam giác đều ABC. Vẽ nửa đường tròn đường kính BC cắt cạnh AB và AC lần lượt tại D và E. Hãy so sánh các cung \(\overset\frown{BD};\overset\frown{BE};\overset\frown{EC}\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

- Đọc dữ kiện đề bài để vẽ hình.

- Chứng minh hai tam giác BOD và EOC là tam giác đều, tính \(\widehat {DOE}\) rồi so sánh các góc suy ra \(\overset\frown{BD}=\overset\frown{BE}=\overset\frown{🌟EC}\)

Lời giải chi tiết

Gọi O là tâm đường tròn đường kính BC. Ta có OB = OD (= R) Vậy tam giác BOD cân tại O

Mà \(\widehat {DBO}\)= 60o nên tam giác BOD đều

Suy ra \(\widehat {DOB}\)= 60o

OE = DC (= R) Vậy tam giác EOC cân tại O

Mà \(\widehat {ECO}\)= 60o nên tam giác EOC đều

Suy ra \(\widehat {EOC}\)= 60o

Ta có \(\widehat {BOD} + \widehat {DOE} + \widehat {EOC} = {180^o}\)

Suy ra 60o + \(\widehat {DOE} + {60^o} = {180^o}\) nên \(\wide🐻hat {DOE} = {60^o}\)

Vì \(\widehat {BOD} = \widehat {DOE} = \widehat {EOC} = {60^o}\) nên sđ\(\overset\frown{BD}\) = sđ\(\overset\frown{BE}\) = sđ\(\overset\frown{EC}={{60}^{o}}\) Vậy \(\overset\frown{BD}=\overset\frown{BE}=\overset\frown{EC}\)

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