let us say X minutes past 6 o clock, the hands are perpendicular. X < 60 .
hours hand moves 360 deg. in 12 hours.
=> 30 deg in 1 hour. => 1/2 deg in 1 minute
minutes hand moves 360 deg in 1 hour
=> 6 deg. in 1 minute
the angle moved by hours hand in X minutes = X/2 deg.
the angle moved by minutes hand in X minutes = 6 X deg.
The difference must be 90 deg. or 270 deg.
6 X - X/2 = 90 deg. => 11/2 X = 90 => X = 180/11 = 16.36 min
6 X - X/2 = 270 deg => X = 540/11 = 49.09 min
So at 6 hrs 16.36 min and at 6 hours 49.09 min, the hands are perpendicular.
