Refer to the diagram below and solve the following intersection problem.  You are given control points Y with coordinates of 100.000N and 100.000E and control point X with coordinates of 120.201N and 181.020E.  Your observe an internal angle at Y of 36 degrees and an internal angle at X of 64 degrees.  Calculate the **EASTING** coordinate of C.  Show your answer to 3 dp and show no units. 




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.

Learn how to calculate the Easting coordinate of point C using trigonometry with detailed step-by-step calculations. This problem involves triangulation, azimuth calculation, and the law of sines. Suitable for advanced high school students or those studying trigonometry.

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