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Giải bài 5 trang 107 sách bài tập toán 9 - Cánh diều tập 2

Cho ngũ giác ABCDE. Chứng minh: AC + AD + BD + BE + EC > AB + BC + CD + DE + EA.

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

Toán - Văn - Anh
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Đề bài

Cho ngũ giác ABCDE. Chứng minh:

🔥AC + AD + BD + BE + EC > AB + BC + CD + DE + EA.

Phương pháp giải - Xem chi tiết

Áp dụng bất đẳng thức trong tam giác là một định lý phát biểu rằng trong một tam giác, chiều dài của một cạnh phải nhỏ hơn tổng, nhưng lớn hơn hiệu của hai cạnh còn lại.

Lời giải chi tiết

AF + FE > AE (trong tam giác AEF);

AJ + JB > AB (trong tam giác ABJ);

BI + IC > BC (trong tam giác BCI);

CH + HD > CD (trong tam giác CDH);

GE + GD > ED (trong tam giác GDE).

Do đó, ta có:

ꦍAF + FE + AJ + JB + BI + IC + CH + HD + GE + GD > AE + AB + BC + CD + ED. (1)

Mặt khác:

🌌(AF + GD) + (JB + FE) + (AJ + IC) + (BI + HD)  + (EG + CH) < AD + BE + AC + BD + EC. 

💜Hay AF + FE + AJ + JB + BI + IC + CH + HD + GE + GD < AB + BC + CD + DE + EA. (2)

Từ (1) và (2) suy ra:

🦩AC + AD + BD + BE + EC > AB + BC + CD + DE + EA.

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Tham Gia Group Dành Cho Lớp 9 - Ôn Thi Vào Lớp 10 Miễn Phí

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