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Giải bài 8 trang 90 vở thực hành Toán 9 tập 2

Cho tam giác nhọn ABC cân tại đỉnh A. Đường tròn đường kính BC cắt các cạnh AB, AC của tam giác ABC lần lượt tại F và E. a) Cho BE cắt CF tại H. Chứng minh rằng AH vuông góc với BC. b) Chứng minh rằng EF song song với BC.

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

Toán - Văn - Anh
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Đề bài

Cho tam giác nhọn ABC cân tại đỉnh A. Đường tròn đường kính BC cắt các cạnh AB, AC của tam giác ABC lần lượt tại F và E. a) Cho BE cắt CF tại H. Chứng minh rằng AH vuông góc với BC. b) Chứng minh rằng EF song song với BC.

Phương pháp giải - Xem chi tiết

a) Chứng minh \(BE \bot AC,CF \bot AB\), suy ra H là trực tâm tam giác ABC nên AH vuông góc với BC. b) Chứng minh \(\widehat {EFC} = \widehat {EBC}\), \(\widehat {EBC} = {90^o} - \widehat {ECB} = {90^o} - \widehat {FBC} = \widehat {FCB}\) nên \(\widehat {EFC} = \widehat {FCB}\), suy ra EF//BC.

Lời giải chi tiết

a) Gọi O là đường tròn đường kính BC. Vì \(\widehat {BEC}\) và \(\widehat {CFB}\) là các góc nội tiếp của (O) cùng chắn nửa đường tròn nên \(\widehat {BEC} = \widehat {CFB} = {90^o}\). Suy ra \(BE \bot AC,CF \bot AB\). Do đó H là trực tâm của tam giác ABC. Vì vậy AH vuông góc với BC. b) Vì \(\widehat {EFC}\) và \(\widehat {EBC}\) là các góc nội tiếp của (O) cùng chắn  nên \(\widehat {EFC} = \widehat {EBC}\) (1) Mặt khác, tam giác ABC cân tại A và các tam giác BCF, CBE lần lượt vuông tại F và E nên \(\widehat {EBC} = {90^o} - \widehat {ECB} = {90^o} - \widehat {FBC} = \widehat {FCB}\). (2) Từ (1) và (2) ta suy ra \(\widehat {EFC} = \widehat {FCB}\). Do đó EF//BC (hai góc ở vị trí so le trong)

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

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