gaokao 2018 Q18

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

As shown in the figure, quadrilateral $A B C D$ is a square, $E$ and $F$ are the midpoints of $A D$ and $B C$ respectively. Using $D F$ as the fold line, fold $\triangle D F C$ so that point $C$ reaches position $P$, with $P F \perp B F$.
(1) Prove: plane $P E F \perp$ plane $A B F D$;
(2) Find the sine of the angle between $D P$ and plane $A B F D$.

(1) From the given conditions, $B F \perp P F$ and $B F \perp E F$, so $B F \perp$ plane $P E F$. Since $B F \subset$ plane $A B F D$, we have plane $P E F \perp$ plane $A B F D$.
(2) Let $P H \perp E F$ with foot $H$. From (1), $P H \perp$ plane $A B F D$. Taking $H$ as the origin, the direction of $\overrightarrow { H F }$ as the positive $y$-axis, and $| \overrightarrow { B F } |$ as the unit length, establish a spatial rectangular coordinate system. From (1), $D E \perp P E$. Since $D P = 2$ and $D E = 1$, we have $P E = \sqrt { 3 }$. Also, $P F = 1$ and $E F = 2$, so $P E \perp P F$. We obtain $P H = \frac { \sqrt { 3 } } { 2 }$ and $E H = \frac { 3 } { 2 }$. Then $H ( 0,0,0 )$, $P \left( 0,0 , \frac { \sqrt { 3 } } { 2 } \right)$, $D \left( -1 , -\frac { 3 } { 2 } , 0 \right)$, $\overrightarrow { D P } = \left( 1 , \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$.

As shown in the figure, quadrilateral $A B C D$ is a square, $E$ and $F$ are the midpoints of $A D$ and $B C$ respectively. Using $D F$ as the fold line, fold $\triangle D F C$ so that point $C$ reaches position $P$, with $P F \perp B F$.

(1) Prove: plane $P E F \perp$ plane $A B F D$;

(2) Find the sine of the angle between $D P$ and plane $A B F D$.

Paper Questions

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