2024年数学高考大一轮复习第八章 培优课 §8.8　 空间距离及立体几何中的探索性问题

§8.8　空间距离及立体几何中的探索性问题

题型一　空间距离

例1 (2022·济宁模拟)如图，在三棱柱ABC－A1B1C1中，AB⊥平面BCC1B1，BC＝AB＝AA1＝2，BC1＝2，M为线段AB上的动点．

(1)证明：BC1⊥CM；

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

(2)若E为A1C1的中点，求点A1到平面BCE的距离．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

思维升华 求点面距一般有以下三种方法．

(1)作点到面的垂线，求点到垂足的距离；

(2)等体积法；

(3)向量法.

跟踪训练1 (1)(2023·枣庄模拟)在长方体ABCD－A1B1C1D1中，AA1＝AB＝2，AD＝1，点F，G分别是AB，CC1的中点，则△D1GF的面积为________．

(2)如图所示，在长方体ABCD－A1B1C1D1中，AD＝AA1＝1，AB＝2，点E在棱AB上移动．

①证明：D1E⊥A1D；

②当E为AB的中点时，求点E到平面ACD1的距离．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

题型二　立体几何中的探索性问题

例2 (2022·常德模拟)如图，三棱柱ABC－A1B1C1的底面是等边三角形，平面ABB1A1⊥平面ABC，A1B⊥AB，AC＝2，∠A1AB＝60°，O为AC的中点．

(1)求证：AC⊥平面A1BO；

(2)试问线段CC1上是否存在点P，使得二面角P－OB－A1的余弦值为，若存在，请计算的值；若不存在，请说明理由．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

思维升华 (1)对于存在判断型问题的求解，应先假设存在，把要成立的结论当作条件，据此列方程或方程组，把“是否存在”问题转化为“点的坐标是否有解，是否有规定范围内的解”等．

(2)对于位置探究型问题，通常借助向量，引进参数，综合已知和结论列出等式，解出参数．

跟踪训练2 如图，四棱锥S－ABCD的底面是正方形，每条侧棱的长都是底面边长的倍，P为侧棱SD上的点．

(1)求证：AC⊥SD；

(2)若SD⊥平面PAC，求二面角P－AC－D的大小；

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

(3)在(2)的条件下，侧棱SC上是否存在一点E，使得BE∥平面PAC.若存在，求SE∶EC的值；若不存在，请说明理由．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________