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Giải bài 8 trang 57 sách bài tập toán 8 - Chân trời sáng tạo

Chứng minh rằng trong một tứ giác, tổng độ dài hai đường chéo lớn hơn nửa chu vi tứ giác đó.

ꦬTổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Chứng minh rằng trong một tứ giác, tổng độ dài hai đường chéo lớn hơn nửa chu vi tứ giác đó.

Phương pháp giải - Xem chi tiết

+ Sử dụng kiến thức về đường chéo trong tứ giác: Trong tứ giác, đường chéo là đoạn thẳng nối hai đỉnh đối nhau. + Sử dụng kiến thức về bất đẳng thức trong tam giác để chứng minh: Trong một tam giác, tổng độ dài hai cạnh tam giác lớn hơn độ dài cạnh còn lại.

Lời giải chi tiết

Vẽ tứ giác ABCD có I là giao điểm của hai đường chéo AC và BD.Theo bất đẳng thức trong tam giác, ta có:\(IA + IB > AB\), \(IB + IC > BC\), \(IC + ID > CB\), \(IA + ID > AD\)Do đó: \(2\left( {IA + IB + IC + ID} \right) > AB + BC + CD + DA\)Hay \(2\left( {AC + BD} \right) > AB + BC + CD + DA\)Do đó: \(AC + BD > \frac{{AB + BC + CD + DA}}{2}\) (đpcm)

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Tham Gia Group Dành Cho Lớp 8 Chia Sẻ, Trao Đổi Tài Liệu Miễn Phí

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