正三棱柱底面边长为2,二面角D-BC-A为45度,求四面体D-ABC的体积(要过程)
\u6b63\u4e09\u68f1\u67f1ABC-A1B1C1\u7684\u5e95\u9762\u6b63\u4e09\u89d2\u5f62\u8fb9\u957f\u662f2,D\u662fCC1\u7684\u4e2d\u70b9,\u76f4\u7ebfAD\u4e0e\u4fa7\u9762BB1C1C\u6240\u6210\u89d245\u5ea6,\u6c42\u4e8c\u9762\u89d2A-BD-C...\u8fc7A\u505aB\u2014C1\u9762\u7684\u5782\u7ebfAE\u3002\u77e5\u89d2ADE=45\u00b0\uff0c\u89e3\u5f97AE=\u6839\u53f73\uff0cDE=AE\uff1bCD=\u6839\u53f72\uff0c\u6545CC1=2\u6839\u53f72\uff1b\u8fc7A\u4f5cBD\u5782\u7ebfAF\uff0c\u6c42\u89e3\u5f97AF=\u6839\u53f7\uff0810/3\uff09\uff1btanAFE=AE/AF\uff1b\u8fc7D\u4f5c\u4e2d\u7ebfDG\uff0c\u6c42\u5f97DG=\u6839\u53f75\uff0cC\u5230\u8fb9DG\u8ddd\u79bb=\u6839\u53f7\uff086/5\uff09
arccos(\u221a21/7)
\u53d6BC\u8fb9\u4e2d\u70b9D\uff0c\u8fde\u63a5AD\u3001A1D\uff0c\u6240\u6c42\u89d2\u5373\u4e3a\u89d2ADA1\uff0c\u4e14\u4e09\u89d2\u5f62ADA1\u4e3a\u76f4\u89d2\u4e09\u89d2\u5f62\uff0c\u7531\u5404\u8fb9\u957f\u5ea6\u7b97\u5f97AD=\u221a3,A1D=\u221a7,\u6bd4\u4e00\u4e0b\u5c31\u51fa\u6765\u4e86\u3002
过点A做BC的垂线,锤足为E,连接DE
因为角AED为45度,角DAB为90度,所以角ADB为45度
所以AD=AB
AB=根号(2^2-1^2)=根号3
所以AD=根号3
三角形ABC的面积为2×根号3×1/2=根号3
四面体D-ABC的体积为:根号3×根号3×1/3=1
取BC中点O 连结AO DO
...
二面角D-BC-A为45度
所以角ACD=45度
等腰Rt三角形AOD中
AD=AO=根号3
V=Sh=根号3*根号3=3
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