TNT2005418
幼苗
共回答了18个问题采纳率:94.4% 举报
【解法1】本题主要考查直线和平面垂直、直线与平面所成的角、二面角等基础知识,考查空间想象能力、运算能力和推理论证能力.
(Ⅰ)∵PA⊥底面ABC,∴PA⊥BC.又
![](https://img.yulucn.com/upload/7/4e/74e0406a933a4392010d453a1cd62172_thumb.jpg)
,∴AC⊥BC.∴BC⊥平面PAC.…………
(Ⅱ)∵D为PB的中点,DE//BC,
∴
![](https://img.yulucn.com/upload/2/c3/2c3ce520f39bd4ba09b05f1c43e0b423_thumb.jpg)
,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,∵PA⊥底面ABC,∴PA⊥AB,又PA=AB,
∴△ABP为等腰直角三角形,∴
![](https://img.yulucn.com/upload/6/be/6be01e72505a6ef4734acaff2b9b1cbd_thumb.jpg)
,
∴在Rt△ABC中,
![](https://img.yulucn.com/upload/c/fd/cfd066b345a1933c63cb831231e034d0_thumb.jpg)
,∴
![](https://img.yulucn.com/upload/d/36/d36fa9871009b3b2fbb9519284b2b657_thumb.jpg)
.
∴在Rt△ADE中,
![](https://img.yulucn.com/upload/4/d4/4d4adfe1a063f82d7ab849f432f182b2_thumb.jpg)
,
∴
![](https://img.yulucn.com/upload/c/a0/ca0b4032b7e305bc2ce24690a786e637_thumb.jpg)
2 与平面
![](https://img.yulucn.com/upload/8/7a/87a4ddf1881764148fa73f9fb1df748d_thumb.jpg)
所成的角的余弦值为
![](https://img.yulucn.com/upload/4/3b/43b85f432623cde99464167e9e6b5584_thumb.jpg)
.…………
(Ⅲ)∵AE//BC,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
![](https://img.yulucn.com/upload/f/38/f38fcd672cd630c5a5bdc9053e69f786_thumb.jpg)
平面PAC,PE
![](https://img.yulucn.com/upload/f/38/f38fcd672cd630c5a5bdc9053e69f786_thumb.jpg)
平面PAC,∴DE⊥AE,DE⊥PE,
∴∠AEP为二面角
![](https://img.yulucn.com/upload/c/a0/ca0b4032b7e305bc2ce24690a786e637_thumb.jpg)
5 的平面角,
∵PA⊥底面ABC,∴PA⊥AC,∴
![](https://img.yulucn.com/upload/2/00/200abaa38bf7c895e7ee872d09e36bff_thumb.jpg)
.
∴在棱PC上存在一点E,使得AE⊥PC,这时
![](https://img.yulucn.com/upload/6/92/69200a329afa993b5d85be9e8c472ec5_thumb.jpg)
,
故存在点E使得二面角
![](https://img.yulucn.com/upload/c/a0/ca0b4032b7e305bc2ce24690a786e637_thumb.jpg)
5 是直二面角.…………
【解法2】如图,以A为原煤点建立空间直角坐标系
![](https://img.yulucn.com/upload/6/c9/6c98874514ffe103cec0374dd425e20b_thumb.jpg)
,
设
![](https://img.yulucn.com/upload/4/3e/43e7803839d180808567248974e08929_thumb.jpg)
,由已知可得
![](https://img.yulucn.com/upload/1/9b/19b83565970b40281759adce7ea5e576_thumb.jpg)
.
(Ⅰ)∵
![](https://img.yulucn.com/upload/2/40/2404cbead7136be9c22e8032762c6167_thumb.jpg)
,∴
![](https://img.yulucn.com/upload/9/8f/98f0dffbae87a0407668482f9ed5708c_thumb.jpg)
,∴BC⊥AP.
又∵
![](https://img.yulucn.com/upload/7/4e/74e0406a933a4392010d453a1cd62172_thumb.jpg)
,∴BC⊥AC,∴BC⊥平面PAC.…………
(Ⅱ)∵D为PB的中点,DE//BC,∴E为PC的中点, ∴
![](https://img.yulucn.com/upload/2/2f/22fef39eb79c6b85a3fb97c02d94691a_thumb.jpg)
,
∴又由(Ⅰ)知,BC⊥平面PAC,∴∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,
∵
![](https://img.yulucn.com/upload/7/5c/75cc98e33b642facc50a8e2ae4af4b55_thumb.jpg)
,∴
![](https://img.yulucn.com/upload/2/33/23396bf5a5321a96210cfb12df960a66_thumb.jpg)
.
∴
![](https://img.yulucn.com/upload/c/a0/ca0b4032b7e305bc2ce24690a786e637_thumb.jpg)
2 与平面
![](https://img.yulucn.com/upload/8/7a/87a4ddf1881764148fa73f9fb1df748d_thumb.jpg)
所成的角的余弦值为
1年前
6