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Câu 20 trang 55 SGK Hình học 11 Nâng cao

Cho tứ diện ABCD và ba điểm P, Q, R lần lượt nằm trên ba cạnh AB, CD, BC. Hãy xác định giao điểm S của mp(PQR) với cạnh AD nếu:
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Cho tứ diện ABCD và ba điểm P, Q, R lần lượt nằm trên ba cạnh AB, CD, BC. Hãy xác định giao điểm S của mp(PQR) với cạnh AD nếu:

LG a

PR // AC

Phương pháp giải:

- Tìm giao tuyến của (PQR) với (ACD). Sử dụng tính chất: Nếu hai mặt phẳng chứa hai đường thẳng song song thì chúng cắt nhau theo giao tuyến song song với đường thẳng đã cho. - Tìm giao điểm S của AD với giao tuyến trên.

Lời giải chi tiết:

Trường hợp PR // AC

Ta có: \(\left\{ \begin{array}{l}
PR \subset \left( {PQR} \right)\\
AC \subset \left( {ACD} \right)\\
PR//AC\\
Q \in \left( {PQR} \right) \cap \left( {ACD} \right)
♐\end{array} \right. \) \(\Rightarrow \left( {PQR} \right) \cap \left( {ACD} \right) = Qt//AC\)

Trong (ACD), gọi S = Qt ∩ AD thì S = AD ∩ (PQR).

LG b

PR cắt AC

Lời giải chi tiết:

Trường hợp PR cắt AC

Trong (ABC), gọi I = PR ∩ AC

\( \Rightarrow \left\{ \begin{array}{l}
I \in AC \subset \left( {ACD} \right)\\
I \in PR \subset \left( {PQR} \right)
🌱\end{array} \right.\)\( \Rightarrow I \in \left( {ACD} \right) \cap \left( {PQR} \right)\)

Mà \( Q\in \left( {ACD} \right) \cap \left( {PQR} \right)\) ⇒ (PQR) ∩ (ACD) = QI Trong mp(ACD) ta có S = QI ∩ AD thì S = AD ∩ (PQR).

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