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Giải bài 15 trang 100 sách bài tập toán 9 - Chân trời sáng tạo tập 1

Cho tam giác ABC cân tại A, (widehat A < {90^o}). Vẽ đường tròn đường kính AB cắt BC và AC lần lượt tại D và E. Chứng minh rằng: a) (Delta DBE) là tam giác cân. b) (widehat {CBE} = frac{1}{2}widehat {BAC})

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

Toán - Văn - Anh
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Đề bài

Cho tam giác ABC cân tại A, \(\widehat A < {90^o}\). Vẽ đường tròn đường kính AB cắt BC và AC lần lượt tại D và E. Chứng minh rằng: a) \(\Delta DBE\) là tam giác cân. b) \(\widehat {CBE} = \frac{1}{2}\widehat {BAC}\)

Phương pháp giải - Xem chi tiết

Chứng minh DE = DB suy ra \(\Delta DBE\) là tam giác cân. Hai góc nội tiếp cùng chắn một cung thì bằng nhau.

Lời giải chi tiết

a) Ta có D, E cùng nằm trên đường tròn đường kính AB nên \(\widehat {ADB} = \widehat {AEB} = {90^o}\) hay \(AD \bot BC\) và \(BE \bot AC\).

Mà tam giác AB𓃲C cân tại A nên D là trung điểm BC nên DE = DB = DC. Vậy tam giác BDE cân tại D.

b) Ta có AD là tia 🔴phân giác của \(\widehat {CAB}\), nên \(\wideha⭕t {BAD} = \widehat {CAD} = \frac{1}{2}\widehat {CAB}\).

Mặt khác \(\widehat{CBE}=\widehat{♊DBE}=\widehat{EAD}=\f♒rac{1}{2}sđ\overset\frown{DE}\).

Suy ra \(\widehat {CBE} = \widehat {🔜BAD} = \frac{1}{2}\widehat {BAC}\).

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

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