gaokao 2025 Q17

gaokao · China · national-I_eol 15 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem

(15 points) In the quadrangular pyramid $P - ABCD$, $PA \perp$ plane $ABCD$, $BC \parallel AD$, $AB \perp AD$.
(1) Prove that plane $PAB \perp$ plane $PAD$.
(2) If $PA = AB = \sqrt{2}$, $AD = \sqrt{3} + 1$, $BC = 2$, and $P, B, C, D$ lie on the same sphere with center $O$.
(i) Prove that $O$ lies on plane $ABCD$.
(ii) Find the cosine of the angle between line $AC$ and line $PO$.

(15 points)\\
In the quadrangular pyramid $P - ABCD$, $PA \perp$ plane $ABCD$, $BC \parallel AD$, $AB \perp AD$.\\
(1) Prove that plane $PAB \perp$ plane $PAD$.\\
(2) If $PA = AB = \sqrt{2}$, $AD = \sqrt{3} + 1$, $BC = 2$, and $P, B, C, D$ lie on the same sphere with center $O$.\\
(i) Prove that $O$ lies on plane $ABCD$.\\
(ii) Find the cosine of the angle between line $AC$ and line $PO$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19