To solve this, we need to break down the problem step by step:

1. Find the azimuth from Point 1 to Point 2

To do this, we first compute the difference in northing and easting coordinates between the two points and use the inverse tangent to calculate the azimuth.

Given coordinates:

• Point 1: $N_1 = 10123.56, E_1 = 10156.92$
• Point 2: $N_2 = 9905.56, E_2 = 9923.57$

We calculate: $\Delta N = N_2 - N_1 = 9905.56 - 10123.56 = -218.00$ $\Delta E = E_2 - E_1 = 9923.57 - 10156.92 = -233.35$

The azimuth (angle from north) from Point 1 to Point 2 is given by: $\text{Azimuth}_{1\rightarrow 2} = \arctan\left(\frac{\Delta E}{\Delta N}\right)$

Let's compute the azimuth.The azimuth from Point 1 to Point 2 is approximately $226.95^\circ$.

2. Add the given angle to the azimuth

Next, we add the clockwise horizontal angle of $312^\circ \, 46' \, 23''$ to this azimuth. First, we need to convert this angle to decimal degrees.

The angle is: $312^\circ + \frac{46}{60} + \frac{23}{3600} \approx 312.77306^\circ$

Now, we add this to the azimuth: $\text{New Azimuth} = 226.95^\circ + 312.77306^\circ = 539.72306^\circ$

Since azimuths are measured in the range $0^\circ$ to $360^\circ$, we subtract 360 if the result exceeds 360. Let's compute this.The new azimuth, after adding the angle, is approximately $179.72^\circ$.

3. Compute the coordinates of the new point

We now calculate the coordinates of a point that is 309.92 feet from Point 1 at this azimuth. To do this, we use the following formulas for the northing and easting: $\Delta N = 309.92 \times \cos(\text{New Azimuth})$ $\Delta E = 309.92 \times \sin(\text{New Azimuth})$ We will compute these values and add them to the coordinates of Point 1. The azimuth must be converted to radians for this calculation.The coordinates of the new point, located 309.92 feet from Point 1 at an azimuth of $179.72^\circ$, are approximately:

• Northing: $9813.64$
• Easting: $10158.43$

Would you like any further details or clarification on this process?

Here are 5 related questions to expand on this:

1. How do you convert an azimuth to a bearing?
2. What is the significance of using radians versus degrees in trigonometry?
3. How do you convert GPS coordinates to UTM (Universal Transverse Mercator)?
4. How can you calculate the distance between two geographic coordinates?
5. What are the different types of angles used in surveying?

Tip: Always ensure your angle calculations remain within the correct range for azimuths (0° to 360°) when adding or subtracting angles.