# Finding Bearings

I have a trigonometry problem that I can't figure out. It goes like this:

Points M and P are the same distance from a third point, O. The bearing of M from O is o38 degrees and the bearing of P from O is 152 degrees. I know the bearings add up to 190 and the answer to the multiple choice question is D: between 180 and 270, but what I can't figure out is how to get the bearing from P to M. Can anyone help?

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I think "the bearing of M from O is $38$ degrees" means if you draw a ray due East from O and put a point Q on it then you have to rotate the segment OQ $38$ degrees counterclockwise to get it to lie along the segment OM. And I think the question of the bearing from P to M is, through what angle must we rotate a segment extending due East from P to get it to lie along the segment from P to M.

If I have all that wrong, then ignore what follows.

Draw a diagram with all these points and lines. Angle POM is $152-38=114$ degrees. Triangle POM is isosceles, so angle PMO is $33$ degrees. Let the line due East from P cross OM at W. Angle PWO equals angle WOX (vertical angles), so it's $38$ degrees. So angle PWM is $142$ degrees. That makes angle MPW $5$ degrees, and that's the bearing of M from P.

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Thank you so much! You are a great help! –  user61474 Feb 8 '13 at 12:16
If you find it serves your purposes, you can "accept" it by clicking in the little check mark next to it. –  Gerry Myerson Feb 8 '13 at 12:32
Seems good, except I remember bearing to refer to clockwise angle from due north (or direction of motion if you are in a ship). However whether you take East or North, counterclockwise or clockwise as your convention, the construction is similar and I think the angle in the new reference would be also same... –  Macavity Feb 8 '13 at 13:00