Cho tứ diện ABCD có BCD là tam giác đều cạnh a. Gọi I là trung điểm của AB. Mp(P) qua I song song với (BCD). Thiết diện của tứ diện cắt bởi mp(P) có diện tích là:
\(\frac{{{a}^{2}}\sqrt{3}}{4}\)
\(\frac{{{a}^{2}}\sqrt{3}}{8}\)
\(\frac{{{a}^{2}}\sqrt{3}}{12}\)
\(\frac{{{a}^{2}}\sqrt{3}}{16}\)
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Gọi H và K lần lượt là trung điểm của AC và AD ta có:
\(\left\{ \begin{array}{l}I \in \left( {ABC} \right) \cap \left( \alpha \right)\\\left( \alpha \right)\parallel BC \subset \left( {ABC} \right)\\IH\parallel BC\end{array} \right. \Rightarrow \left( {ABC} \right) \cap \left( \alpha \right) = IH\)
Tương tự ta chứng minh được \(\left( \alpha \right)\cap \left( ACD \right)=HK\,\,;\,\,\left( \alpha \right)\cap \left( ABD \right)=IK\)
Vậy thiết diện của hình chóp cắt bởi mặt phẳng \(\left( \alpha \right)\) qua I và song song với (BCD) là tam giác IHK.
Ta có: IH, HK, IK lần lượt là đường trung bình của các tam giác ABC, ACD, ABD.
\(IH=\frac{1}{2}BC=\frac{1}{2}a,HK=\frac{1}{2}CD=\frac{1}{2}a,IK=\frac{1}{2}BD=\frac{1}{2}a\Rightarrow IH=HK=IK=\frac{a}{2}\Rightarrow \Delta IHK\(đều cạnh \(\frac{a}{2}.\)
Vậy \({{S}_{\Delta IHK}}={{\left( \frac{a}{2} \right)}^{2}}\frac{\sqrt{3}}{4}=\frac{{{a}^{2}}\sqrt{3}}{16}.\)
Chọn D.
Bộ Đề Kiểm Tra Giữa Học Kì I - Toán 11 - Các Trường THPT Trên Toàn Quốc không chỉ giúp học sinh củng cố kiến thức đã học mà còn rèn luyện khả năng tư duy logic và giải quyết vấn đề nhanh chóng, chính xác. Đề thi bao gồm các dạng bài tập đa dạng để các em chuẩn bị tốt cho kỳ thi khác.
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Sóng biếc theo làn hơi gợn tí,
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Nhiệt độ ảnh hướng đến sự trao đổi nước và chất dinh dưỡng ở thực vật ra sao?
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