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Giải bài tập 4.11 trang 78 SGK Toán 9 tập 1 - Kết nối tri thức

Tính các góc của hình thoi có hai đường chéo dài (2sqrt 3 )và 2.

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

Toán - Văn - Anh
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Đề bài

Tính các góc của hình thoi có hai đường chéo dài \(2\sqrt 3 \) và 2.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

෴Hình thoi có hai đường chéo cắt nhau tại trung điểm của mỗi đường, đường chéo là phân giác của góc hình thoi, hai đường chéo vuông góc với nhau, từ đó ta có tam giác vuông và các số đo các cạnh của tam giác đó, để tính tỉ số lượng giác, giải các góc của hình thoi

Chú ý: Các góc đối của hình thoi bằng nhau.

Lời giải chi tiết

Theo đề bài ta có: \(AC = 2\sqrt3;\; BD = 2\). Gọi giao điểm của hai đường chéo AC và BD là O. Suy ra \(AO = OC = \sqrt3\); \(BO = OD = 1\). Xét tam giác AOB vuông tại O, ta có: \(tan ABO = \frac{AO}{BO}=\frac{\sqrt3}{1} =\sqrt3\) Suy ra \(\widehat{ABO} = 60^\circ\) suy ra \(\widehat{BAO} = 90^\circ - 60^\circ = 30^\circ\) Theo tính chất hình thoi, ta có: \(\widehat{ABC} = 2\widehat{ABO} = 120^\circ\) \(\widehat{BAD} = 2\widehat{BAO} = 60^\circ\) Vậy hình thoi có các góc là \(120^\circ\) và \(60^\circ\).

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