(本小题满分14分)如图, 在直三棱柱ABC-A1B1C1中,AC=3,BC=4, ,AA1=4,点D是AB的中点. (
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(本小题满分14分) 如图, 在直三棱柱ABC-A1B1C1中,AC=3,BC=4, ![]() ![]() (1)求证:AC⊥BC1; (2)求 ![]() (3)求二面角 ![]() |
![(本小题满分14分)如图, 在直三棱柱ABC-A1B1C1中,AC=3,BC=4, ,AA1=4,点D是AB的中点. (](/uploads/image/z/14460604-52-4.jpg?t=%EF%BC%88%E6%9C%AC%E5%B0%8F%E9%A2%98%E6%BB%A1%E5%88%8614%E5%88%86%EF%BC%89%E5%A6%82%E5%9B%BE%2C+%E5%9C%A8%E7%9B%B4%E4%B8%89%E6%A3%B1%E6%9F%B1ABC%EF%BC%8DA1B1C1%E4%B8%AD%EF%BC%8CAC%EF%BC%9D3%EF%BC%8CBC%EF%BC%9D4%EF%BC%8C+%EF%BC%8CAA1%EF%BC%9D4%EF%BC%8C%E7%82%B9D%E6%98%AFAB%E7%9A%84%E4%B8%AD%E7%82%B9%EF%BC%8E+%EF%BC%88)
(1)证明:直三棱柱ABC-A1B1C1,底面三边长AC=3,
BC=4,AB=5,
∴ AC⊥BC, …………2分
又 AC⊥C1 C,
∴ AC⊥平面BCC1;
∴ AC⊥BC1 …………4分
(2)
…………8分
(3)解法一:取
中点
,过
作
于
,连接
。
是
中点,
∴
∴
平面
,又
∴
∴
,又
∴
平面
∴
∴
是二面角
的平面角…………10分
AC=3,BC=4,AA1=4,
∴
,
∴
,
∴二面角
的余弦值为
…………14分
解法二:以
分别为
轴建立如图所示空间直角坐标系,
AC=3,BC=4,AA1=4,
∴
,
,
,
∴
,
平面
的法向量
,
设平面
的法向量
,
则
,
BC=4,AB=5,
![](http://img.wesiedu.com/upload/2/1a/21a2502caaa235f0d18a0d74def30604.jpg)
![](http://img.wesiedu.com/upload/2/13/213ae2a9001ef8c4d08242e92858834f.jpg)
∴ AC⊥BC, …………2分
又 AC⊥C1 C,
![](http://img.wesiedu.com/upload/3/84/38456d664508ddc30dfd2051688ae992.jpg)
∴ AC⊥平面BCC1;
∴ AC⊥BC1 …………4分
(2)
![](http://img.wesiedu.com/upload/a/36/a36ec3e6a79b2705b6c29ee24aae53e5.jpg)
![](http://img.wesiedu.com/upload/9/b2/9b21b2666c8d528a3d82cf1bb2f61aed.jpg)
(3)解法一:取
![](http://img.wesiedu.com/upload/e/34/e34055f8b939ef1fd2d289553d911133.jpg)
![](http://img.wesiedu.com/upload/2/b1/2b169be20174d77b84f05ee13fd2e16c.jpg)
![](http://img.wesiedu.com/upload/b/72/b729d83492fae63a88d794c7ea108609.jpg)
![](http://img.wesiedu.com/upload/a/f5/af52bb7a8238c36a6a4a95ed302dfcb0.jpg)
![](http://img.wesiedu.com/upload/4/7d/47d3466a4ca5b33e612928e48ecd1210.jpg)
![](http://img.wesiedu.com/upload/e/bb/ebb88b38b332b66830263741eeb644bb.jpg)
![](http://img.wesiedu.com/upload/c/67/c67e40cfb850832e481eb467203fbda8.jpg)
![](http://img.wesiedu.com/upload/2/1a/21a2502caaa235f0d18a0d74def30604.jpg)
![](http://img.wesiedu.com/upload/b/72/b729d83492fae63a88d794c7ea108609.jpg)
![](http://img.wesiedu.com/upload/e/78/e787539bd147c8bb92c3c38123568bcf.jpg)
∴
![](http://img.wesiedu.com/upload/9/b6/9b635533898ce3a27bb640980c186b01.jpg)
∴
![](http://img.wesiedu.com/upload/1/da/1dafe2c20b818650522c989c54331952.jpg)
![](http://img.wesiedu.com/upload/6/79/679a7e4451d6f5da2ac1827bb0c1d5a1.jpg)
![](http://img.wesiedu.com/upload/7/1c/71c40d5b802ca1f956e2e08238ab43ba.jpg)
∴
![](http://img.wesiedu.com/upload/8/75/8750b041058383a93a5bb215b905cd1b.jpg)
∴
![](http://img.wesiedu.com/upload/c/e0/ce03053a1ca0339627d14f40848e5484.jpg)
![](http://img.wesiedu.com/upload/0/ba/0ba176fd780be750902a8c258a79e3c1.jpg)
∴
![](http://img.wesiedu.com/upload/6/ef/6ef5a517abd645d9a921453cdc6ff961.jpg)
![](http://img.wesiedu.com/upload/8/c2/8c2f28bf77a26ab34e8e7a8c1c369a80.jpg)
∴
![](http://img.wesiedu.com/upload/c/8f/c8f37d74b0687edda85e89ea581ee75a.jpg)
∴
![](http://img.wesiedu.com/upload/9/f9/9f9ccb0f87007aa80f6ecc8e03ee7891.jpg)
![](http://img.wesiedu.com/upload/0/2e/02e2c598b64ba0731ce341333bd7ef4f.jpg)
![](http://img.wesiedu.com/upload/2/1a/21a2502caaa235f0d18a0d74def30604.jpg)
∴
![](http://img.wesiedu.com/upload/c/ff/cff68081192cdc8e61e08633c1a445a7.jpg)
![](http://img.wesiedu.com/upload/5/a6/5a658d010fae2f1b32d258b8092cf91e.jpg)
![](http://img.wesiedu.com/upload/a/9d/a9d32e5f1274ce453ae74555b7867d58.jpg)
![](http://img.wesiedu.com/upload/1/c2/1c2c050cb00842d3d2fb257241474b87.jpg)
![](http://img.wesiedu.com/upload/0/2e/02e2c598b64ba0731ce341333bd7ef4f.jpg)
![](http://img.wesiedu.com/upload/5/0d/50dfc7b13c77a64b6afafe0d0d2ab8cc.jpg)
解法二:以
![](http://img.wesiedu.com/upload/b/53/b530fe5d3e2a599fcbca80151a076eb7.jpg)
![](http://img.wesiedu.com/upload/0/da/0da3c0b0cc3c3146300aaed6b0236855.jpg)
![](http://img.wesiedu.com/upload/9/2e/92e8e241736aace56be79a51622845a8.jpg)
![](http://img.wesiedu.com/upload/2/1a/21a2502caaa235f0d18a0d74def30604.jpg)
∴
![](http://img.wesiedu.com/upload/e/10/e10bfb536ec7abe2838149e500fa74df.jpg)
![](http://img.wesiedu.com/upload/3/04/3047916af7b2f0c9a3ffd46b9cc29bb5.jpg)
![](http://img.wesiedu.com/upload/5/c3/5c3dc59d53c65b4460f3b64419a1d1b8.jpg)
∴
![](http://img.wesiedu.com/upload/b/6c/b6cc257310c71e44ea76ba4752c21af2.jpg)
![](http://img.wesiedu.com/upload/7/72/77236511a991c2cbc886d4f727b360ea.jpg)
平面
![](http://img.wesiedu.com/upload/5/e6/5e666befc129fde3003c0b488f7d002b.jpg)
![](http://img.wesiedu.com/upload/4/58/458b197d748965cdfe4f782668bfdc23.jpg)
设平面
![](http://img.wesiedu.com/upload/b/2e/b2e3f0df6b2a3fc535b3a0e26e04dfd2.jpg)
![](http://img.wesiedu.com/upload/3/c6/3c6145e1cf4cc797298f3f50cbd093d5.jpg)
则
![](http://img.wesiedu.com/upload/1/12/1122812ff098334f35f34f2252f05a98.jpg)
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