Question

已知四棱柱ABCD一A1B1C1D1的侧棱AA1垂直于底面,底面ABCD为直角梯形,AD∥BC,AB⊥BC,AD=AA1

Answer 1

（1）取CD的中点F,连接EF,则EF是△DA1C的中位线,∴EF∥A1C．∵A1C⊂面A1BC,

EF在面A1BC 外,∴EF∥平面A1BC．

（2）∵底面ABCD为直角梯形,AD∥BC,AB⊥BC,AD=AA1=2,AB=BC=1,∴AC=CD=√2

由勾股定理得 AC⊥CD．再由AA1垂直于底面得,AA1⊥CD．而AC和AA1是平面A1ACC1内的两条相交直线,

∴CD⊥平面A1ACC1．

矩形A1ACC1的面积等于 AA1×AC=2×√2=2√2,四棱锥D-A1ACC1的高 CD=√2

四棱锥D-A1ACC1的体积为1/3AA1×AC×CD=4/3