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Giải bài tập 6 trang 74 SGK Toán 9 tập 2 - Chân trời sáng tạo

Cho tam giác ABC vuông tại A. Lấy điểm M bất kì trên đoạn AC, đường tròn đường kính CM cắt hai đường thẳng BM và BC lần lượt tại D và N. Chứng minh rằng: a) Tứ giác ABCD nội tiếp; b) Các đường thẳng AB, MN, CD cùng đi qua một điểm.

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 vuông tại A. Lấy điểm M bất kì trên đoạn AC, đường tròn đường kính CM cắt hai đường thẳng BM và BC lần lượt tại D và N. Chứng minh rằng: a) Tứ giác ABCD nội tiếp; b) Các đường thẳng AB, MN, CD cùng đi qua một điểm.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

-  Đọc kĩ dữ liệu để vẽ hình.

-  Dựa vào góc nội tiếp chắn nửa đường tròn bằng 90o và trong một tứ giác nội tiếp, tổng số đo hai góc đối nhau bằng 180o để suy ra tứ giác ABCD nội tiếp.

-  Chứng minh AB, MN, CD là ba đường cao của tam giác MBC suy ra các đường thẳng AB, MN, CD cùng đi qua điểm E.

Lời giải chi tiết

a) Ta có \(\widehat {MDC} = {90^o}\) (góc nội tiếp chắn nửa đường tròn đường kính MC) Xét tam giác ABC có \(\widehat {BAC} = {90^o}\) nên là tam giác vuông tại A, do đó A, B, C thuộc đường tròn đường kính BC. Xét tam giác BCD có \(\widehat {BDC} = {90^o}\) nên là tam giác vuông tại D, do đó B, C, D thuộc đường tròn đường kính BC. Suy ra bốn điểm A, B, C, D thuộc đường tròn đường kính BC hay tứ giác ABCD nội tiếp. b) Ta có \(\widehat {MNC} = {90^o}\) (góc nội tiếp chắn nửa đường tròn đường kính MC) Xét tam giác MBC có AB \( \bot \) MC, CD \( \bot \) BM, MN \( \bot \) BC. Nên AB, MN, CD là ba đường cao của tam giác MBC Vậy các đường thẳng AB, MN, CD cùng đi qua điểm E.

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