7/10 Rule

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Use this rule when the first bearing between the object and the bow is 2 pts (or 22.5), and the second bearing is 4 pts (or 45).
 
What it tells you is that the distance run between the two bearings will be equal to the distance to the object when from the second position. You also know that 0.7 multiplied by the distance run is equal to the distance to the object when abeam, and is also the distance along the track until the object will be abeam. You can also find the ETA for when the object will be abeam by dividing 0.7xDR (in nm) by your speed (in knots), and then adding the result to the time of the second observation.
 
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. Also, because the second triangle is a 45-45-90, we can find the length of the legs using the pythagorean theorum. Knowing the distances, we can then divide by speed to get time.

Back to Special Case Bearings

7-10_rule.jpg

Example:
Question: At 0800 a light is spotted two points to port. At 0830, the same light is spotted four points to port. Our speed is 10 kts, when will the light be abeam? 
 
Solution:
-First, draw out the picture like above.
-Second, find the distance run between the two observations.
     10kt/hr x 0.5 hr = 5 nm
     DR = 5 nm.
-If we know that the distance from the point of the second observation to the point at which the light is abeam is equal to 0.7xDR, we can substitute 5 for DR and solve for the distance until the light is abeam.
     0.7x5 = 3.5nm
-Knowing that we are traveling at 10 kts, how long will it take us to cover those 3.5nm?
     3.5/10 = 0.35 hrs = 21 min.
-Add this to 0830 to find the time at which the light will be abeam:
     0830 + 0021 = 0851