To solve this problem, we'll use the concept of triangulation to determine the coordinates of point C. Given the two control points $Y$ and $X$, their coordinates, and the internal angles at these points, we can calculate the position of point $C$ by applying trigonometry.

Step-by-Step Solution:

1. Define Known Quantities:

   • Control Point $Y$: $(100.000, 100.000)$
   • Control Point $X$: $(120.201, 181.020)$
   • Angle at $Y$: $36^\circ$
   • Angle at $X$: $61^\circ$

2. Determine the Azimuth from Point $Y$ to Point $X$: The azimuth (or bearing) from point $Y$ to point $X$ is the angle that line $YX$ makes with the positive East direction. To compute this, we use the formula: $\text{Azimuth}_{YX} = \tan^{-1} \left( \frac{\Delta N}{\Delta E} \right)$ where:

   • $\Delta N = 120.201 - 100.000 = 20.201$
   • $\Delta E = 181.020 - 100.000 = 81.020$

   Let's calculate the azimuth:

$\text{Azimuth}_{YX} = \tan^{-1} \left( \frac{20.201}{81.020} \right)$

3. Determine the Angles to Point $C$:

   • Angle from $Y$ to $C$: Azimuth from $Y$ to $X$ $+ 36^\circ$.
   • Angle from $X$ to $C$: Azimuth from $X$ to $Y$ $+ 61^\circ$.

4. Calculate the Position of Point $C$: We need to solve the triangle formed by points $X, Y,$ and $C$ using the law of sines.

Let's compute these calculations in Python.The Easting coordinate of point $C$ is 168.222 (rounded to three decimal places).

Would you like a detailed explanation of the steps, or do you have any other questions?

Related Questions:

1. How do you calculate the Northing coordinate of point $C$?
2. What is the distance between points $X$ and $C$?
3. How do you convert between azimuths and bearings?
4. What is the significance of using the law of sines in triangulation problems?
5. How can you find the coordinates of point $C$ if additional points are introduced?

Tip: When working with angles and coordinates, it's crucial to keep track of the direction and reference point to avoid errors in your calculations.