Question

已知三棱柱ABC-A1B1C1的侧棱与底面边长都相等,A1在底面ABC上的射影D为BC的中点,则异面直线AB与CC1所成

Answer 1

设A1在底面ABC上的射影为D,连接AD.侧棱长为X

过D作DE⊥AB,交AB于E,连接A1E

AD=Xsin60°=根号3/2*X

A1D=根号下[X^2-(根号3/2*X)^2]=X/2

DE=X/2*sin60°=根号3/4*X

在直角三角形A1DE中：A1E=根号下[（X/2)^2+(根号3/4*X）^2]=根号7/4*X

AE=根号下【X^2-(根号7/4*X)^2]=3/4*X

cos∠A1ADB=AE/A1A=3/4*X/X=3/4