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Giải bài 31 trang 86 SBT toán 10 - Cánh diều

Cho đường tròn tâm O và dây cung BC không đi qua O.

Tổng hợp đề thi học kì 2 lớp 10 tất cả cá🙈c môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Cho đường tròn tâm O và dây cung BC không đi qua O. Điểm A chuyển động trên cung lớn BC của đường tròn sao cho tam giác ABC nhọn. Gọi H là trực tâm của tam giác ABC. Chꦚứng minh rằng \(\overri🐷ghtarrow {AH} \) có độ dài không đổi.

Phương pháp giải - Xem chi tiết

Chứng minh AH không đổi

Bước 1: Kẻ đường kính BD của đường tròn (O)

Bước 2: Chứng minh ADCH là hình bình hành (dấu hiệu tứ giác có các🥃 cặp cạnh song💯 song)

Bước 3: Từ giả thiết bước 2 suy ra AH = DC rồi kết luận

Lời giải chi tiết

Kẻ đường kính BD của đường tròn (O)

Ta có:

\(\widehat {BAD}\) là góc nội tiếp chắn nửa đường tròn nên \(\widehat {BAD}\) = 900 \( \Rightarrow AD \bot AB\) (1)

Mặt khác, \(CH \bot AB\) (giả thiết) (2)

Từ (1) và (2) suy ra AD // CH (3)

Chứng minh tương tự ta được AH // CD (4)

Từ (3) và (4) suy ra ADCH là hình bình hành \( \Rightarrow \)AH = DC

Mà DC không đổi nên AH không đổi

\( \Rightarrow \)\(\overrightarrow {AH} \) có độ dài không đổi 

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Ph/hs Tham Gia Nhóm Để Cập Nhật Điểm Thi, Điểm Chuẩn Miễn Phí

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