E. Example Intersections
Two different intersection types are solved using triangle and arcbased methods to demonstrate the different computation process. Additional decimal places will be carried in computations to minimize rounding errors.
1. Distancedistance
Given the information on the diagram, determine the coordinates of point 101.


Figure E1 
Step (1) For both methods is to inverse along the base line 3020
a. Trianglebased method
Step (2) Compute angle at 30 by Law of Cosines. 

Step (3) Compute direction from 30 to 101. 

Step (4) Perform a forward computation from 30 to 101. 
Math Check: Compute coordinates from 20.
Step (1) Compute angle at 20 by Law of Sines. 

Step (2) Compute direction from 20 to 101. 

Step (3) Perform a forward computation from 20 to 101. Both coordinates check. 
b. Arcbased method
Step (2) Set up and solve Equations D6 through D9.
Step (3) Use Equations D10 and D11 to compute the two intersection points
Step (4) Of the two, select the appropriate intersection point.
Point 101 is located southwest of the base line.
Point  North  East  From base line 
101_{1}  5077.015  1363.991  north east 
101_{2}  4574.617  1085.956  south west 
The correct intersection point is 101_{2}: (4754.617 ft N, 1085.956 ft E), same as the trianglebased solution.
2. Directiondistance
Figure xx2 is a diagram of a base line and an intersection point.
Figure E2 
Coordinates of the base line end points are:
Point  North (ft)  East (ft) 
J  1419.51  3511.69 
K  2056.64  2341.36 
The azimuth from point J to point T is 329°46'45"; the distance from point K to point T is 738.15 ft.
What are the coordinates of point T?
Step (1) Inverse the base line from point K to point J
a. Trianglebased method
To get point T's coordinates, we'll perform a forward computation from K.
Step (2) Compute the angle at J
Step (3) Use Law of Sines to compute angle at T
Remember that we have to check the angle returned by the Law of Sines because it will always be <90°.
From the diagram, T is greater than 90° so it must be subtracted from 180°
Step (4) Compute angle at K from the Angle Condition
Step (5) Compute azimuth from point K to point T
Step (6) Forward compute from point K
The math check is to compute the coordinates from point J
Step (1) Compute distance from point J using Law of Sines
Step (2) Forward compute from point J
b. Arcbased method
Because the direction can intersect the arc at two points, we'll extend the diagram a bit to visualize the possibilities, Figure XXXX.
Figure E3 
Step (2) Set up and solve Equations D12 through D15.
Step (3) Pick the correct distance and compute the coordinates
From Figure E3, the correct distance d_{Q} is the shorter one: 878.915,. Using that distance, perform a forward computation
The coordinates of point T are the same as those from the trianglebased method.