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Giải bài tập 10 trang 82 SGK Toán 9 tập 2 - Chân trời sáng tạo

Cho tam giác nhọn ABC có đường cao AH (H ( in ) BC) và nội tiếp đường tròn tâm O có đường kính AM (hình 6). Chứng minh (widehat {OAC} = widehat {BAH}).

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 có đường cao AH (H \( \in \) BC) và nội tiếp đường tròn tâm O có đường kính AM (hình 6). Chứng minh \(\widehat {OAC} = \widehat {BAH}\).

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Dựa vào góc nội tiếp chắn nửa đường tròn bằng 90o để chứng minh\(\widehat {OAC} + \widehat {OCM} ﷽= {90^o}\).

Theo hình vẽ ta chứng minh \(\widehat {OAC} = \widehat {OCA} = {90^o} - \widehat {OCM} = \widehat {BAH}\)

Lời giải chi tiết

OA = OC = R nên \(\Delta \)OAC cân tại O. Vì \(\widehat {ACM}\) là góc nội tiếp chắn cung AM, AM là đường kính đường tròn (O). Suy ra \(\widehat {ACM} = {90^o}\) hay \(\widehat {OAC} + \widehat {OCM} = {90^o}\) suy ra \(\widehat {OAC} = \widehat {OCA} = {90^o} - \widehat {OCM}\)  Vì OC = OM = R nên tam giác OMC cân tại O suy ra \(\widehat {OCM} = \widehat {OMC}\). Do đó \(\widehat {OAC} = {90^o} - \widehat {OMC}\) Vì \(\widehat {OMC}\) và \(\widehat B\) cùng là góc nội tiếp chắn cung AC nhỏ nên \(\widehat {OAC} = {90^o} - \widehat {B} = \widehat {BAH}\) (tổng ba góc trong của tam giác).

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

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