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Câu 5 trang 125 SGK Hình học 11 Nâng cao

Cho hình hộp ABCD.A’B’C’D’. Điểm M nằm giữa A và D, điểm N nằm giữa C và C’ sao cho
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Đề bài

Cho hình hộp ABCD.A’B’C’D’. Điểm M nằm giữa A và D, điểm N nằm giữa C và C’ sao cho \({{AM} \over {MD}} = {{CN} \over {NC'}}\) a. Chứng minh rằng đường thẳng MN song song với mp(ACB’)  b. Xác định thiết diện của hình hộp khi cắt bởi mặt phẳng đi qua MN và song song với mp(ACB’)

Lời giải chi tiết

a. \({{AM} \over {MD}} = {{CN} \over {NC'}} \Rightarrow {{AM} \over {CN}} = {{MD} \over {NC'}} = {{AD} \over {CC'}}\)theo định lí Ta-lét đảo thì MN song song mp(P), ở đó (P) song song với AC và DC’Mặt khác DC’ // AB’. Vậy MN // (ACB’)b. Kẻ MK // AC (K ϵ CD); kẻ NI // CB’ (I ϵ C’B’); kẻ IJ // A’C’ (J ϵ A’B’) ;Kẻ JE // AB’ (E ϵ AA’).Thiết diện là lục giác MKNIJE.

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