Question

A boundary survey is being conducted for a client’s parcel of land. You find and accept a concrete monument for a boundary corner and determine its position to have X and Y coordinates of 3496.45 ft & 9798.03 feet, respectively (Corner A). You also find & accept an iron rod as a corner and determine its position to have X and Y coordinates of 5272.92 ft & 10161.36 feet, respectively (Corner C). You determine that you should replace a lost corner (Corner B) between these accepted monuments using record distances provided on a 1995 survey map. The record distance from Corner A to B is 1405.91’. The record distance from Corner C to B is 1121.30’. What are the coordinates the field crew should use to reset corner B? HINT: You will get 2 solutions using distance-distance intersection techniques. Calculate both solutions but chose the solution that is northeasterly from Corner A & northwesterly from Corner C as your final solution. Also, use Civil 3D to check your answer. Show screenshot of your steps. Hint: by plotting the given coordinates and then using circles with radii, we can determine the intersection points.

Answer 1

So, I'll use coordinate geometry and algebra to find out the two possible solutions as shown:
Given, Corner A (3496.45, 9798.03) and Corner C (5272.92, 10161.36)

Let us assume the x coordinate of the corner B is x and y coordinate of the corner B is y.
We know that, as per the distance formula, distance between two points in a 2-D plane is given by

Where, x₁ and x₂ are the x coordinates of the first and second points respectively and y₁ and y₂ are the y coordinates of the first and second points respectively.
Using the above formula we can find out two equations in x and y and solve them to find their values.

The distance between corners A and B is 1405.91 ft. Using the above formula,

...................... (1)

Also, the distance between corners B and C is 1121.3 ft.

..........................(2)

Solving the above two equations,
We get two pairs of solutions,
1. x = 4408.82 & y = 10871.09
2. x = 4753.18 & y = 9167.79

But the problem specifies that the corner B is North Wasterly from corner A which means that the Y coordinate of corner B must be greater than the y coordinate of corner A and the x coordinate of corner B must be greater than the x coordinate of corner A.

Also, the problem states that the corner B is North Westerly from corner C which means that the Y coordinate of corner B must be greater than the y coordinate of corner C and the x coordinate of corner B must be lesser than the x coordinate of corner C.

The first solution satisfies these conditions.
Hence, the required corner B has coordinates (4408.82, 10871.09)

I do not have Civil 3D in my system right now. But it won't be a problem for you. Mark the given locations, A and C with their coordinates in Civil 3D and draw two circles of radii 1405.91 units and 1121.3 unit keeping the centres at A and C respectively. These two circles will intersect at two locations. Those two locations will be our solutions which will reduce to one, final solution after applying further given two conditions of direction.

Thank you! Upvotes are appreciated.