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Giải bài 9.30 trang 56 sách bài tập toán 9 - Kết nối tri thức tập 2

Cho hình thoi ABCD có AC=8cm, BD=4cm. Gọi M, N, P, Q lần lượt là trung điểm của AB, BC, CD, DA. Chứng minh rằng tứ giác MNPQ nội tiếp một đường tròn và tìm bán kính của đường tròn đó.

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

Toán - Văn - Anh
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Đề bài

Cho hình thoi ABCD có AC=8cm, BD=4cm. Gọi M, N, P, Q lần lượt là trung điểm của AB, BC, CD, DA. Chứng minh rằng tứ giác MNPQ nội tiếp một đường tròn và tìm bán kính của đường tròn đó.

Phương pháp giải - Xem chi tiết

+ Gọi O là giao điểm của AC và BD. + Tính được OA, OB, OC, OD và chỉ ra AC vuông góc với BD tại O. + Chứng minh M là tâm đường tròn ngoại tiếp tam giác OAB, từ đó tính được MO. + Chứng minh được \(MO = NO = PO = OQ\) nên tứ giác MNPQ nội tiếp đường tròn (O) bán kính MO.

Lời giải chi tiết

Gọi O là giao điểm của AC và BD. Vì ABCD là hình thoi nên \(OA = OC = \frac{1}{2}AC = 4cm,OB = OD = \frac{1}{2}BD = 2cm\) và AC vuông góc với BD tại O. Áp dụng định lí Pythagore vào tam giác ABO vuông tại O có: \(A{B^2} = O{A^2} + O{B^2} = 20\) nên \(AB = 2\sqrt 5 cm\). Do đó, \(AB = BC = CD = DA = 2\sqrt 5 cm\). Vì M là trung điểm của AB, tam giác AOB vuông tại O nên M là tâm đường tròn ngoại tiếp tam giác OAB và \(MO = MA = MB = \frac{{AB}}{2} = \sqrt 5 cm\). Tương tự ta có: \(MO = NO = PO = OQ = \sqrt 5 cm\). Vậy tứ giác MNPQ nội tiếp đường tròn (O) bán kính \(\sqrt 5 cm\).

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