名校
1 . 在正三棱台
中,
是边长为
的等边三角形,且
.已知
,
,
,
分别是线段
,
的中点,当直线
上一动点
在射线
上时,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/45fdfe37-a174-4faf-9513-6a3beda731bf.png?resizew=247)
(1)证明:
平面
;
(2)求直线
与平面
所成角的正弦值;
(3)连接
,
,已知点
在平面
投影是
,平面
是一个分别以
,
作为
,
轴的复平面,
.当
时,请直接写出
的虚部(不要求写出过程).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c6e8a07dcdeec1d196fd31a616d25d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed78b13131d8ce1c65091b6b259a1082.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d84cdf3d579140b8e5b6f9f4efcc23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efac9930e80e002a537bc0a6da526866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48c79f3aa20d5255d03a498afafbf727.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfc386b9493102a587025523dc69ccba.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/45fdfe37-a174-4faf-9513-6a3beda731bf.png?resizew=247)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f6d39135a2f8472d66ea00eda3b13ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b94e97d085cea077cb82a0b7d2f523e.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
(3)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abaeba15f3abdd877bc701af52c5cd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4819c39c281427826e1b3f7a4c2b720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4819c39c281427826e1b3f7a4c2b720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d822262ff00915910e5b87d81ad1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71b63d2504bd3ecce8c10560b142356f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/904cb7508ea31d4d1e3b39b594decfad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45224f7eac9d0cef64bf28d93e7721a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
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2021-11-22更新
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