1. **State the problem:** We have a lot shaped as a polygon with corners 1, 2, 3, and 4. We know azimuths and distances for lines 2-3, 3-4, and 3-1, but distances and azimuths for lines 1-2 and 4-1 are missing. We want to analyze the geometry and possibly find those missing values. 2. **Convert azimuths to bearings or directions:** Azimuth is measured clockwise from north. - Line 2-3 azimuth: $326^\circ 00'$ equals $326^\circ$ - Line 3-4 azimuth: $88^\circ 00'$ equals $88^\circ$ - Line 3-1 azimuth: $112^\circ 00'$ equals $112^\circ$ 3. **Calculate coordinates relative to point 3:** Assume point 3 at origin $(0,0)$. - For line 2-3 (from 2 to 3): distance $116.40$ m, azimuth $326^\circ$. To find point 2 coordinates: $$x_2 = -116.40 \sin(326^\circ)= -116.40 \sin(326^\circ)$$ $$y_2 = -116.40 \cos(326^\circ) = -116.40 \cos(326^\circ)$$ - For line 3-4: distance $174.40$ m, azimuth $88^\circ$: $$x_4 = 174.40 \sin(88^\circ)$$ $$y_4 = 174.40 \cos(88^\circ)$$ - For line 3-1: distance $216.60$ m, azimuth $112^\circ$: $$x_1 = 216.60 \sin(112^\circ)$$ $$y_1 = 216.60 \cos(112^\circ)$$ 4. **Calculate the coordinates numerically:** - $\sin(326^\circ) = \sin(360^\circ - 34^\circ) = -\sin(34^\circ) \approx -0.5592$ - $\cos(326^\circ) = \cos(360^\circ - 34^\circ) = \cos(34^\circ) \approx 0.8290$ Thus, $$x_2 = -116.40 \times -0.5592 = 65.08$$ $$y_2 = -116.40 \times 0.8290 = -96.47$$ - $\sin(88^\circ) \approx 0.9994$ - $\cos(88^\circ) \approx 0.0349$ Thus, $$x_4 = 174.40 \times 0.9994 = 174.30$$ $$y_4 = 174.40 \times 0.0349 = 6.09$$ - $\sin(112^\circ) = \sin(180^\circ - 68^\circ) = \sin(68^\circ) \approx 0.9272$ - $\cos(112^\circ) = -\cos(68^\circ) = -0.3746$ Thus, $$x_1 = 216.60 \times 0.9272 = 200.87$$ $$y_1 = 216.60 \times -0.3746 = -81.16$$ 5. **Find distances and azimuths of missing lines:** - Line 1-2 distance: $$d_{1-2} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(65.08 - 200.87)^2 + (-96.47 +81.16)^2}$$ $$= \sqrt{(-135.79)^2 + (-15.31)^2} = \sqrt{18437 + 234} = \sqrt{18671} \approx 136.58 \text{ m}$$ - Line 4-1 distance: $$d_{4-1} = \sqrt{(x_1 - x_4)^2 + (y_1 - y_4)^2} = \sqrt{(200.87 - 174.30)^2 + (-81.16 - 6.09)^2}$$ $$= \sqrt{(26.57)^2 + (-87.25)^2} = \sqrt{706 + 7618} = \sqrt{8324} \approx 91.26 \text{ m}$$ - Azimuth of line 1-2 (from 1 to 2): $$\theta_{1-2} = \arctan \frac{x_2 - x_1}{y_2 - y_1} = \arctan \frac{65.08 - 200.87}{-96.47 + 81.16} = \arctan \frac{-135.79}{-15.31}$$ Both numerator and denominator negative, so angle in third quadrant. Calculate magnitude: $$|\arctan| = \arctan(135.79/15.31) \approx \arctan(8.87) \approx 83.57^\circ$$ So azimuth = $180^\circ + 83.57^\circ = 263.57^\circ$ - Azimuth of line 4-1 (from 4 to 1): $$\theta_{4-1} = \arctan \frac{x_1 - x_4}{y_1 - y_4} = \arctan \frac{200.87 - 174.30}{-81.16 - 6.09} = \arctan \frac{26.57}{-87.25}$$ Numerator positive, denominator negative means angle in second quadrant. Calculate magnitude: $$|\arctan| = \arctan(26.57/87.25) \approx \arctan(0.3046) \approx 16.92^\circ$$ Azimuth = $180^\circ - 16.92^\circ = 163.08^\circ$ **Final answers:** - Missing distance 1-2 is approximately $136.58$ m. - Missing distance 4-1 is approximately $91.26$ m. - Azimuth 1-2 is approximately $264^\circ$. - Azimuth 4-1 is approximately $163^\circ$. These complete the lot's data for all sides.