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Giải bài 7.3 trang 26 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho tứ diện \(ABCD\), gọi \(M\) và \(N\) lần lượt là trung điểm của \(AC\) và \(BD\).

♚Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Cho tứ diện \(ABCD\), gọi \(M\) và \(N\) lần lượt là trung điểm của \(AC\) và \(BD\). Biết \(MN = a\sqrt 3 ;AB = 2\sqrt 2 a\) và \(CD = 2a\). Chứng minh rằng đường thẳng \(AB\) vuông góc với đường thẳng \(CD\).

Phương pháp giải - Xem chi tiết

Chứng minh góc giữa đường thẳng \(AB\) và \(CD\)  bằng \({90^ \circ }\) + Bước 1: Tính góc giữa hai đường thẳng \(AB\) và \(CD\) bằng \({90^ \circ }\) + Bước 2: Kết luận  đường thẳng \(AB\) vuông góc với đường thẳng \(CD\). Chú ý sử dụng định lý đảo Pytago để chứng minh tam giác là tam giác vuông

Lời giải chi tiết

Lấy \(K\) là trung điểm của cạnh \(BC\), ta có: \(NK\) và \(MK\) lần lượt là đường trung bình của tam giác \(BCD\) và tam giác \(ABC\) nên \(NK = a,MK = a\sqrt 2 \).Do đó, \(M{N^2} = 3{a^2} = N{K^2} + M{K^2}\) suy ra tam giác \(MNK\) vuông tại \(K\), hay \(MK \bot NK\), mà \(MK//AB\) và \(NK//CD\) nên \(\left( {AB,CD} \right) = \left( {MK,NK} \right) = {90^ \circ }\), hay \(AB \bot CD\).

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