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Giải Bài 72 trang 90 sách bài tập toán 7 - Cánh diều

Chứng minh: Nếu một tam giác có hai đường trung tuyến bằng nhau thì tam giác đó là tam giác cân.

𒀰Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Cánh diều

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Đề bài

Chứng minh: Nếu một tam giác có hai đường trung tuyến bằng nhau thì tam giác đó là tam giác cân.

Phương pháp giải - Xem chi tiết

Giả sử tam giác ABC có hai đường trung tuyến BM và CN bằng nhau chứng minh tam giác ABC cân tại A.

Lời giải chi tiết

 

Tam giác ABC có hai trung tuyến BM và CN bằng nhau.Gọi G là giao điểm của BM và CN.Theo tính chất trọng tâm tam giác có: \(BG = \frac{2}{3}BM\)và \(CG = \frac{2}{3}CN\).Vì BM = CN nên BG = CG.Suy ra tam giác BGC cân tại G.Do đó \(\widehat {GBC} = \widehat {GCB}\) (hai góc ở đáy).Xét ∆MBC và ∆NCB có:BC là cạnh chung,\(\widehat {MBC} = \widehat {NCB}\) (do \(\widehat {GBC} = \widehat {GCB}\))MB = NC (giả thiết)Do đó ∆MBC = ∆NCB (c.g.c)Suy ra \(\widehat {MCB} = \widehat {NBC}\) (hai góc tương ứng).Khi đó tam giác ABC cân tại A.Vậy nếu một tam giác có hai đường trung tuyến bằng nhau thì tam giác đó là tam giác cân.

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