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Giải bài 4.24 trang 63 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho tứ diện ABCD. Gọi G và H lần lượt là trọng tâm của hai tam giác ABC và ACD. Chứng minh rằng GH//(BCD)

Tổng hợp đề thi học kì 2 lớp 11🍸 tất c🌳ả các môn - Kết nối tri thức

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Đề bài

Cho tứ diện ABCD. Gọi G và H lần lượt là trọng tâm của hai tam giác ABC và ACD. Chứng minh rằng GH//(BCD)

Phương pháp giải - Xem chi tiết

Nếu đường thẳng a không nằm trong mặt phẳng (P) và a song song với một đường thẳng nằm trong mặt phẳng (P) thì a song song với (P)

Lời giải chi tiết

Gọi E, F lần lượt là trung điểm của các cạnh BC, CD. Vì G là trọng tâm của tam giác ABC nên A, G, E thẳng hàng và \(\frac{{AG}}{{AE}} = \frac{2}{3}\)Tương tự ta có A, H, F thẳng hàng và \(\frac{{AH}}{{AF}} = \frac{2}{3}.\)Do đó, \(\frac{{AG}}{{AE}} = \frac{{AH}}{{AF}}\)Trong tam giác AEF có: \(\frac{{AG}}{{AE}} = \frac{{AH}}{{AF}}\), theo định lí Thalès đảo ta có GH//EF, mà \(EF \subset \left( {BCD} \right)\) nên GH//(BCD) 

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