Let's break down the problem step by step. --- ## **Given Data** - **Coordinates of point 16:** \( X_{16} = 2227.819 \) m \( Y_{16} = 3335.821 \) m - **From 16 to 17:** - **Azimuth:** \( 166^\circ 55' 48'' \) - **Distance:** \( 769.745 \) m - **From 16 to 18:** - **Azimuth:** \( 226^\circ 23' 24'' \) - **Distance:** \( 598.532 \) m --- ## **Step 1: Compute Coordinates for Points 17 and 18** ### **Formulas** \[ \Delta X = D \cdot \sin(\theta) \] \[ \Delta Y = D \cdot \cos(\theta) \] Where \( \theta \) is the azimuth (in decimal degrees, converted to radians). --- ### **A. Coordinates for Point 17** #### **Convert Azimuth to Decimal Degrees** \[ 166^\circ 55' 48'' = 166 + \frac{55}{60} + \frac{48}{360} = 166.930^\circ \] #### **Convert to Radians** \[ \theta_{16 \to 17} = 166.930^\circ \times \frac{\pi}{180} = 2.9143 \text{ radians} \] #### **Compute ΔX and ΔY** \[ \Delta X_{16 \to 17} = 769.745 \times \sin(166.930^\circ) \] \[ \Delta Y_{16 \to 17} = 769.745 \times \cos(166.930^\circ) \] \[ \sin(166.930^\circ) \approx .22744 \] \[ \cos(166.930^\circ) \approx -.97379 \] \[ \Delta X_{16 \to 17} = 769.745 \times .22744 = 175.006 \text{ m} \] \[ \Delta Y_{16 \to 17} = 769.745 \times (-.97379) = -749.799 \text{ m} \] \[ X_{17} = X_{16} + \Delta X_{16 \to 17} = 2227.819 + 175.006 = 2402.825 \text{ m} \] \[ Y_{17} = Y_{16} + \Delta Y_{16 \to 17} = 3335.821 - 749.799 = 2586.022 \text{ m} \] --- ### **B. Coordinates for Point 18** \[ 226^\circ 23' 24'' = 226 + \frac{23}{60} + \frac{24}{360} = 226.39^\circ \] \[ \theta_{16 \to 18} = 226.39^\circ \times \frac{\pi}{180} = 3.952 \text{ radians} \] \[ \sin(226.39^\circ) \approx -.6286 \] \[ \cos(226.39^\circ) \approx -.7777 \] \[ \Delta X_{16 \to 18} = 598.532 \times (-.6286) = -376.323 \text{ m} \] \[ \Delta Y_{16 \to 18} = 598.532 \times (-.7777) = -465.251 \text{ m} \] \[ X_{18} = 2227.819 + (-376.323) = 1851.496 \text{ m} \] \[ Y_{18} = 3335.821 + (-465.251) = 287.570 \text{ m} \] --- ## **Step 2: Calculate Azimuth and Distance** ### **From 17 to 18** \[ \Delta X = X_{18} - X_{17} = 1851.496 - 2402.825 = -551.329 \text{ m} \] \[ \Delta Y = Y_{18} - Y_{17} = 287.570 - 2586.022 = 284.548 \text{ m} \] #### **Distance** \[ D_{17 \to 18} = \sqrt{(-551.329)^2 + (284.548)^2} \] \[ D_{17 \to 18} = \sqrt{304963.3 + 81069.6} = \sqrt{386032.9} \approx 621.3 \text{ m} \] #### **Azimuth** \[ \theta = \arctan\left(\frac{\Delta X}{\Delta Y}\right) \] \[ \arctan\left(\frac{-551.329}{284.548}\right) \approx -62.93^\circ \] But this is in the 2nd quadrant (since ΔX negative, ΔY positive): \[ \text{Azimuth} = 180^\circ + (-62.93^\circ) = 117.07^\circ \] But azimuths are measured from North (clockwise): \[ \theta = \arctan2(\Delta X, \Delta Y) \] \[ \theta = \arctan2(-551.329, 284.548) = -62.93^\circ \] Since ΔX negative, ΔY positive, the angle is in the 2nd quadrant (between 90° and 180°): \[ \text{Azimuth} = 360^\circ + (-62.93^\circ) = 297.07^\circ \] But let's check: - ΔX negative, ΔY positive: 2nd quadrant azimuth = 180 - |θ| - θ = 62.93°, so Azimuth = 180 - 62.93 = **117.07°** ### **From 18 to 17** Just reverse ΔX and ΔY: \[ \Delta X = X_{17} - X_{18} = 2402.825 - 1851.496 = 551.329 \text{ m} \] \[ \Delta Y = Y_{17} - Y_{18} = 2586.022 - 287.570 = -284.548 \text{ m} \] \[ \theta = \arctan2(551.329, -284.548) \] This is positive X, negative Y: 4th quadrant. \[ \arctan2(551.329, -284.548) = 117.07^\circ \] So, Azimuth = 360 - 62.93 = **297.07°** --- ## **Step 3: Compare** - **From 17 to 18:** Distance ≈ 621.3 m, Azimuth ≈ 117.07° - **From 18 to 17:** Distance ≈ 621.3 m, Azimuth ≈ 297.07° These azimuths differ by 180°, as expected. --- ## **Final Answers** ### **Coordinates** - \( (X_{17}, Y_{17}) = (2402.825, 2586.022) \) - \( (X_{18}, Y_{18}) = (1851.496, 287.570) \) ### **From 17 to 18** - **Distance:** \( 621.3 \) m - **Azimuth:** \( 117.07^\circ \) ### **From 18 to 17** - **Distance:** \( 621.3 \) m - **Azimuth:** \( 297.07^\circ \) ### **Comparison** - The distances are the same. - The azimuths differ by 180°.