Point 1 N = 10123.56 E = 10156.92 Point 2 N = 9905.56 E = 9923.57

Compute the coordinates of a point located 309.92 ft. from point 1 at a clockwise horizontal angle of 312 d 46’ 23” from a line from 1 to 2.

To solve this problem, we need to:

1. **Compute the azimuth** (bearing) of the line from Point 1 to Point 2.
2. **Adjust the azimuth** by the given angle (312° 46' 23") to find the direction of the new line.
3. **Compute the new coordinates** using the adjusted azimuth and the distance of 309.92 feet.

### Step 1: Compute the Azimuth from Point 1 to Point 2

The azimuth is the angle measured clockwise from the North direction to the line connecting the two points. 

We can calculate it using the formula:

\[
\text{Azimuth} = \arctan\left(\frac{\Delta E}{\Delta N}\right)
\]

where \(\Delta E = E_2 - E_1\) and \(\Delta N = N_2 - N_1\).

Given:
- Point 1: \(N_1 = 10123.56\), \(E_1 = 10156.92\)
- Point 2: \(N_2 = 9905.56\), \(E_2 = 9923.57\)

\[
\Delta N = 9905.56 - 10123.56 = -218.00
\]
\[
\Delta E = 9923.57 - 10156.92 = -233.35
\]

\[
\text{Azimuth} = \arctan\left(\frac{-233.35}{-218.00}\right)
\]

Now, let's calculate this value.

\[
\text{Azimuth} = \arctan\left(1.0704\right) 
\]
\[
\text{Azimuth} \approx 47.73^\circ 
\]

Since both \(\Delta E\) and \(\Delta N\) are negative, the direction is in the third quadrant, so we add 180° to get the correct azimuth:

\[
\text{Azimuth from Point 1 to Point 2} = 227.73^\circ
\]

### Step 2: Adjust the Azimuth by the Given Angle

Next, we add the clockwise horizontal angle (312° 46' 23'') to the azimuth we just calculated.

\[
\text{New Azimuth} = 227.73^\circ + 312.773^\circ 
\]

\[
\text{New Azimuth} = 540.503^\circ 
\]

Since azimuths are generally given in the range of 0° to 360°, subtract 360°:

\[
\text{New Azimuth} = 540.503^\circ - 360^\circ = 180.503^\circ
\]

### Step 3: Compute the New Coordinates

Now that we have the azimuth and the distance, we can compute the new coordinates using:

\[
\Delta N' = d \times \cos(\text{New Azimuth})
\]
\[
\Delta E' = d \times \sin(\text{New Azimuth})
\]

where \(d = 309.92\) ft, and the new azimuth is 180.503°.

Let's calculate \(\Delta N'\) and \(\Delta E'\):

\[
\Delta N' = 309.92 \times \cos(180.503^\circ) 
\]
\[
\Delta E' = 309.92 \times \sin(180.503^\circ) 
\]

### Calculate the New Coordinates

Finally, add these to the coordinates of Point 1:

\[
N_{\text{new}} = N_1 + \Delta N'
\]
\[
E_{\text{new}} = E_1 + \Delta E'
\]

Let’s perform the final calculations for \(\Delta N'\) and \(\Delta E'\).The coordinates of the new point, located 309.92 feet from Point 1 at a clockwise horizontal angle of 312° 46' 23" from the line connecting Point 1 and Point 2, are approximately:

- **Northing (N):** 9813.65
- **Easting (E):** 10154.20

Would you like further details or have any questions about the solution?

### Related Questions:
1. How would the calculation change if the distance was in meters instead of feet?
2. What if the angle was counterclockwise instead of clockwise?
3. How can you compute the inverse direction, i.e., from the new point back to Point 1?
4. How would the azimuth calculation differ if the points were in a different quadrant?
5. How can you perform similar calculations using software like AutoCAD or GIS tools?

### Tip:
Always double-check your azimuth adjustments, especially when working across different quadrants or when the sum exceeds 360°.

Point 1 N = 10123.56 E = 10156.92 Point 2 N = 9905.56 E = 9923.57. Compute the coordinates of a point located 309.92 ft. from point 1 at a clockwise horizontal angle of 312° 46’ 23” from a line from 1 to 2.

Learn how to calculate the coordinates of a point based on azimuth and distance. This example covers key geometry and trigonometry concepts using given Northing and Easting coordinates. Suitable for students in Grades 10-12.

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