Cho hình hộp \(A B C D \cdot A^{\prime} B^{\prime} C^{\prime} D^{\prime}\), trên cạnh \(A A^{\prime}, B B^{\prime}, C C^{\prime}\) lần lượt lấy ba điểm \(M, N, P\) sao cho \(\frac{A M}{A A^{\prime}}=\frac{3}{4}, \frac{B N}{B B^{\prime}}=\frac{1}{2}, \frac{C P}{C C^{\prime}}=\frac{1}{3}\). Biết rằng \((M N P)\) cắt \(D^{\prime} D\) tại \(Q\). Tính tỷ số \(\frac{D^{\prime} Q}{D^{\prime} D}\).
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Ta có \(:\left\{\begin{array}{l}\left(B B^{\prime} C C^{\prime}\right) / /\left(A A^{\prime} D D^{\prime}\right) \\ (M N P) \cap\left(B B^{\prime} C C^{\prime}\right)=N P \Rightarrow N P / / M Q \\ (M N P) \cap\left(A A^{\prime} D D^{\prime}\right)=M Q\end{array}\right.\) (1)
Và \(\left\{\begin{array}{l}(A A B B) / /(C C D D) \\ (M N P) \cap(A A B B)=M N \\ (M N P) \cap(C C D D)=P Q\end{array} \Rightarrow M N / / P Q\right.\) (2)
Từ (1) và (2) suy ra mặt phẳng \((M N P)\) cắt hình hộp \(A B C D \cdot A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) theo thiết diện là hình bình hành \(MNPQ\).
Gọi \(I=A C \cap B D, K=M P \cap N Q\). Dễ dàng có \(I K\) là đường trung bình của hai hình thang \(A C P M\) và \(B D Q N\) nên \(I K=\frac{A M+C P}{2}=\frac{B N+D Q}{2}\) (3)
Suy ra \(A M=\frac{1}{4} A A^{\prime}, B N=\frac{1}{2} B B^{\prime}=\frac{1}{2} A A^{\prime}, C P=\frac{2}{3} C C^{\prime}=\frac{2}{3} A A^{\prime}\).
Do đó (3) \(\Rightarrow D Q=\frac{5}{12} D D^{\prime}\).
Vậy \(\frac{D^{\prime} Q}{D^{\prime} D}=\frac{7}{12}\).
Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2025 – ĐHQG Hà Nội – Đề Số 05 là tài liệu luyện thi toàn diện, giúp học sinh làm quen với cấu trúc và độ khó của kỳ thi ĐGNL theo chuẩn mới. Bộ test gồm đầy đủ 3 phần: Toán học và Xử lý số liệu, Văn học – Ngôn ngữ, và Khoa học/Tiếng Anh, được thiết kế với thời lượng, số câu hỏi và thang điểm đúng theo đề thi chính thức. Các câu hỏi trong bộ test không chỉ kiểm tra kiến thức mà còn rèn luyện tư duy phản biện, lập luận logic và khả năng vận dụng thực tiễn. Đây là tài liệu quan trọng giúp thí sinh tự đánh giá năng lực, luyện tốc độ, nâng cao kỹ năng giải đề và chuẩn bị vững vàng cho kỳ thi Đánh Giá Năng Lực năm 2025 của ĐHQG Hà Nội.
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