Bow and Beam

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Use this rule when the first bearing between the object and the bow is 45, and the second bearing is 90. Though limited in use, it is useful in certain cases.
 
What it tells you is that the distance run between the two bearings will be equal to the distance to the object when it is abeam.
 
This works because, as you can see in the figure at right, the triangle created when drawn out is isoceles, meaning that two of the sides and two of the interior angles are the same. In a triangle, the sum of the three interior angles will be 180. If one angle is a, another is 180-2a, and the last is b, simply solving the equation
180 = a + b + 180 - 2a will show that a = b, and therefore both angles are the same. Because both angles are the same, the sides opposite them will be of equal length.

Back to Special Case Bearings

bow_and_beam.jpg

Example:
Question: You are steering course 090 at 12kts. At 0800 you spot a buoy at 045. At 0815, the same buoy bears 000. How far away is the buoy when abeam?
 
Solution:
-First, find your relative bearings.
   090 - 045 = 45         
   090 - 000 = 90
-Second, find the Distance Run from the first observation to the second.
   0815 - 0800 = 15 min
   15min/60 x 12 kts = 3 nm
-As per the bow and beam rule, the Distance Run is equal to the distance to the object when abeam. Therefore the object is 3nm away.