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2023年高考数学新高考Ⅰ-18

  2023-07-08 11:03:38  

(12分)如图,在正四棱柱$ABCD-A_{1}B_{1}C_{1}D_{1}$中,$AB=2$,$AA_{1}=4$.点$A_{2}$,$B_{2}$,$C_{2}$,$D_{2}$分别在棱$AA_{1}$,$BB_{1}$,$CC_{1}$,$DD_{1}$上,$AA_{2}=1$,$BB_{2}=DD_{2}=2$,$CC_{2}=3$.
(1)证明:$B_{2}C_{2}//A_{2}D_{2}$;
(2)点$P$在棱$BB_{1}$上,当二面角$P-A_{2}C_{2}-D_{2}$为$150^\circ$时,求$B_{2}P$.

分析:(1)建系,根据坐标法及向量共线定理,即可证明;
(2)建系,根据向量法,向量夹角公式,方程思想,即可求解.
解:

(1)证明:根据题意建系如图,则有:
$B_{2}(0$,2,$2)$,$C_{2}(0$,0,$3)$,$A_{2}(2$,2,$1)$,$D_{2}(2$,0,$2)$,
$\therefore$$\overrightarrow{{B}_{2}{C}_{2}}=(0,-2,1)$,$\overrightarrow{{A}_{2}{D}_{2}}=(0,-2,1)$,
$\therefore$$\overrightarrow{{B}_{2}{C}_{2}}=\overrightarrow{{A}_{2}{D}_{2}}$,又$B_{2}$,$C_{2}$,$A_{2}$,$D_{2}$四点不共线,
$\therefore B_{2}C_{2}//A_{2}D_{2}$;
(2)在(1)的坐标系下,可设$P(0$,2,$t)$,$t\in [0$,$4]$,
又由(1)知$C_{2}(0$,0,$3)$,$A_{2}(2$,2,$1)$,$D_{2}(2$,0,$2)$,
$\therefore$$\overrightarrow{{C}_{2}{A}_{2}}=(2,2,-2)$,$\overrightarrow{{C}_{2}P}=(0,2,t-3)$,$\overrightarrow{{A}_{2}{D}_{2}}=(0,-2,1)$,
设平面$PA_{2}C_{2}$的法向量为$\overrightarrow{m}=(x,y,z)$,
则$\left\{\begin{array}{l}{\overrightarrow{m}\cdot \overrightarrow{{C}_{2}{A}_{2}}=2x+2y-2z=0}\\ {\overrightarrow{m}\cdot \overrightarrow{{C}_{2}P}=2y+(t-3)z=0}\end{array}\right.$,取$\overrightarrow{m}=(t-1,3-t,2)$,
设平面$A_{2}C_{2}D_{2}$的法向量为$\overrightarrow{n}=(a,b,c)$,
则$\left\{\begin{array}{l}{\overrightarrow{n}\cdot \overrightarrow{{C}_{2}{A}_{2}}=2a+2b-2c=0}\\ {\overrightarrow{n}\cdot \overrightarrow{{A}_{2}{D}_{2}}=-2b+c=0}\end{array}\right.$,取$\overrightarrow{n}=(1,1,2)$,
$\therefore$根据题意可得$\vert \cos 150^\circ \vert =\vert \cos  < \overrightarrow{m}$,$\overrightarrow{n} > \vert =\dfrac{\vert \overrightarrow{m}\cdot \overrightarrow{n}\vert }{\vert \overrightarrow{m}\vert \vert \overrightarrow{n}\vert }$,
$\therefore$$\dfrac{\sqrt{3}}{2}=\dfrac{6}{\sqrt{(t-1)^{2}+(3-t)^{2}+4}\times \sqrt{6}}$,
$\therefore t^{2}-4t+3=0$,又$t\in [0$,$4]$,
$\therefore$解得$t=1$或$t=3$,
$\therefore P$为$B_{1}B_{2}$的中点或$B_{2}B$的中点,
$\therefore B_{2}P=1$.
点评:本题考查利用向量法证明线线平行,利用向量法求解二面角问题,向量共线定理及向量夹角公式的应用,方程思想,属中档题.

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