作业帮如何证明正方体的体对角线与面对角线所成角为90度?高一学子呵呵.未学摄影定理.求详解.静候佳音.予以无尽感谢.
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/28 20:40:46
如何证明正方体的体对角线与面对角线所成角为90度?高一学子呵呵.未学摄影定理.求详解.静候佳音.予以无尽感谢.
如何证明正方体的体对角线与面对角线所成角为90度?
高一学子呵呵.未学摄影定理.求详解.静候佳音.予以无尽感谢.
如何证明正方体的体对角线与面对角线所成角为90度?高一学子呵呵.未学摄影定理.求详解.静候佳音.予以无尽感谢.
设边长a,那么面对角线为根号2*a,体对角线为根号3*a
把面对角线沿棱方向往中间平移a/2,则必和体对角线有交点,然后连线体对角线的一个端点和平移后的面对角线端点,设为L,它和面对角线的一半和体对角线的一半组成一个三角形
那么:L^2=a^2+(a/2)^2=5/4*a^2
面对角线的一半的平方=(根号2/2)^2=1/2*a^2
体对角线的一半的平方=（根号3/2)^2=3/4*a^2
在这个三角形内,L的平方等于两条对角线的一半的平方的和,可知是直角三角形
因此正方体的体对角线与面对角线所成角为90度
线面垂直啊！
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