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Giải bài 10 trang 37 sách bài tập toán 9 - Cánh diều tập 1

Cho tứ giác ABCD. Chứng minh diện tích của tứ giác ABCD không lớn hơn (frac{{AB.BC + AD.DC}}{2}.)

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

Toán - Văn - Anh
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Đề bài

Cho tứ giác ABCD. Chứng minh diện tích của tứ giác ABCD không lớn hơn \(\frac{{AB.BC + AD.DC}}{2}.\)

Phương pháp giải - Xem chi tiết

Kẻ đường cao CH  và AK. Tính diện tích tam giác ABC và ACD.  Do đó\({S_{ABCD}} = {S_{ABC}} + {S_{ACD}} = \frac{{AB.CH + DC.AK}}{2}\) Kết hợp với điều kiện \(CH \le BC,AK \le AD\), ta được điều phải chứng minh.

Lời giải chi tiết

Kẻ \(CH \bot AB,AK \bot DC(H \in AB,K \in DC)\). Ta có \({S_{ABC}} = \frac{{AB.CH}}{2},{S_{ACD}} = \frac{{DC.AK}}{2}\) Do đó \({S_{ABCD}} = {S_{ABC}} + {S_{ACD}} \\= \frac{{AB.CH}}{2} + \frac{{DC.AK}}{2} = \frac{{AB.CH + DC.AK}}{2}\) Mà \(CH \le BC,AK \le AD\) suy ra \({S_{ABCD}} \le \frac{{AB.BC + AD.DC}}{2}\) Vậy diện tích của tứ giác ABCD không lớn hơn \(\frac{{AB.BC + AD.DC}}{2}.\)

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