gaokao 2018 Q18

gaokao · China · national-I-arts 12 marks Not Maths

As shown in the figure, in parallelogram $A B C M$, $A B = A C = 3$ and $\angle A C M = 90 ^ { \circ }$. Fold $\triangle A C M$ along $AC$ so that point $M$ moves to position $D$, with $A B \perp A D$.
(1) Prove: Plane $A C D \perp$ Plane $A B C$
(2) Let $Q$ be a point on segment $AD$ and $P$ be a point on segment $BC$ such that $B P = D Q = \frac { 2 } { 3 } D A$. Find the volume of the tetrahedron $Q - A B P$.

(1) From the given conditions, $\angle B A C = 90 ^ { \circ }$ and $B A \perp A C$. Since $B A \perp A D$, we have $A B \perp$ Plane $A C D$. Since $A B \subset$ Plane $A B C$, therefore Plane $A C D \perp$ Plane $A B C$.
(2) From the given conditions, $D C = C M = A B = 3$ and $D A = 3 \sqrt { 2 }$. Construct $Q E \perp A C$ with foot $E$. Then $Q E = \frac { 1 } { 3 } D C$. From the given conditions and (1), $D C \perp$ Plane $A B C$, so $Q E \perp$ Plane $A B C$ and $Q E = 1$. Therefore, the volume of tetrahedron $Q - A B P$ is $V _ { Q - A B P } = \frac { 1 } { 3 } \cdot S_{\triangle ABP} \cdot QE$.

As shown in the figure, in parallelogram $A B C M$, $A B = A C = 3$ and $\angle A C M = 90 ^ { \circ }$. Fold $\triangle A C M$ along $AC$ so that point $M$ moves to position $D$, with $A B \perp A D$.\\
(1) Prove: Plane $A C D \perp$ Plane $A B C$\\
(2) Let $Q$ be a point on segment $AD$ and $P$ be a point on segment $BC$ such that $B P = D Q = \frac { 2 } { 3 } D A$. Find the volume of the tetrahedron $Q - A B P$.

Paper Questions

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