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Giải bài 13 trang 100 sách bài tập toán 9 - Chân trời sáng tạo tập 1

Cho tam giác ABC nhọn với các đường cao AA’, BB’, CC’. Chứng minh rằng A’A là tia phân giác của góc (widehat {B'A'C'}).

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

Toán - Văn - Anh
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Đề bài

꧑Cho tam giác ABC nhọn với các đường cao AA’, BB’, CC’. Chứng minh rằng A’A là tia phân giác của góc \(\widehat {B'A'C'}\).

Phương pháp giải - Xem chi tiết

Chứng minh bốn điểm B, A’, H, C’ cùng nằm trên một đường tròn Nếu hai tiếp tuyến của một đường tròn cắt nhau tại một điểm thì tia kẻ từ điểm đó đi qua tâm là tia phân giác của góc tạo bởi hai tiếp tuyến.

Lời giải chi tiết

Gọi H là trực tâm của tam giác ABC.

ཧTa có \(\widehat {BC'H} = \widehat {BA'H} = {90^o}\), nên bốn điểm B, A’, H, C’ cùng nằm trên một đường tròn.

Do đó \(\widehat {HA'C'} = \widehat {HBC'}\).

🧜Chứng minh tương tự, ta cũng có \(\widehat {HA'B'} = \widehat {HCB'}\).

ꦍMà \(\widehat {HBC'} = \widehat {HCB'}\) (cùng phụ với \(\widehat {BAC}\)), nên ta có \(\widehat {C'A'H} = \widehat {B'A'H}\).

🎶Từ đó, ta có A’A là tia phân giác của góc \(\widehat {B'A'C'}\).

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

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