To solve this problem, we will use the new internal angles provided at points $Y$ and $X$ and apply triangulation to determine the coordinates of point $C$.

Updated Known Quantities:

• Control Point $Y$: $(100.000, 100.000)$
• Control Point $X$: $(120.201, 181.020)$
• Angle at $Y$: $36^\circ$
• Angle at $X$: $64^\circ$

Steps to Find the Easting Coordinate of Point $C$:

1. Calculate the Azimuth from Point $Y$ to Point $X$:

   The azimuth (or bearing) from point $Y$ to point $X$ is calculated as: $\text{Azimuth}_{YX} = \tan^{-1} \left( \frac{\Delta E}{\Delta N} \right)$ where:

   • $\Delta N = X_N - Y_N = 120.201 - 100.000 = 20.201$
   • $\Delta E = X_E - Y_E = 181.020 - 100.000 = 81.020$

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

2. Determine the Azimuths to Point $C$:

   • Azimuth from $Y$ to $C$: $\text{Azimuth}_{YX} + 36^\circ$
   • Azimuth from $X$ to $C$: $\text{Azimuth}_{YX} + 180^\circ - 64^\circ$ (because it's on the opposite side)

3. Apply the Law of Sines to Find Distance to Point $C$:

   Let:

   • $\angle C = 180^\circ - 36^\circ - 64^\circ = 80^\circ$

   Use the law of sines to find distances from $Y$ to $C$ ($YC$) and from $X$ to $C$ ($XC$):

$\frac{\text{dist}_{YC}}{\sin(64^\circ)} = \frac{\text{dist}_{XC}}{\sin(36^\circ)} = \frac{\text{dist}_{YX}}{\sin(80^\circ)}$

4. Calculate the Easting Coordinate of Point $C$:

   Let's perform these calculations now.The updated Easting coordinate of point $C$ is 170.658 (rounded to three decimal places).

Would you like further details on the solution, 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 point $X$ and point $C$?
3. How do changes in internal angles affect the position of point $C$?
4. How can the azimuths be used to verify the correct position of point $C$?
5. What are some practical applications of triangulation in real-world scenarios?

Tip: Always double-check the angles and directions when performing triangulation calculations to ensure accuracy.