Inaccessible 82710

Determine the distance between two inaccessible places K, L, if the angles KAL=62°10", LAB=41°23", KBL=66°34", and LBA were measured from points A, B, which are 870 m apart = 34°52".
Thank you.

Correct answer:

x =  749.853 m

Step-by-step explanation:

$$\begin{gathered} AB=870\text{m} \\ LBA=34°52^{\prime }=34°+\frac{52}{60}°=34.8667°\doteq 34.8667 \\ LAB=21°23^{\prime }=21°+\frac{23}{60}°=21.3833°\doteq 21.3833 \\ KAB=52°40^{\prime }=52°+\frac{40}{60}°=52.6667°\doteq 52.6667 \\ KBA=81°15^{\prime }=81°+\frac{15}{60}°=81.25°=81.25 \\ \angle ALB=180-\angle LAB-\angle LBA=180-21.3833-34.8667=\frac{495}{4}=123.75{}^{\circ } \\ \angle AKB=180-\angle KBA-\angle KAB=180-81.25-52.6667=\frac{553}{12}\doteq 46.0833{}^{\circ } \\ LA:AB=\sin LBA:\sin ALB \\ LA=AB\cdot \frac{\sin LBA}{\sin ALB}=AB\cdot \frac{\sin 34.866666666667°}{\sin 123.75°}=870\cdot \frac{\sin 34.866666666667°}{\sin 123.75°}=870\cdot \frac{0.571669}{0.83147}=598.15982\text{m} \\ KB=AB\cdot \frac{\sin KAB}{\sin AKB}=AB\cdot \frac{\sin 52.666666666667°}{\sin 46.083333333333°}=870\cdot \frac{\sin 52.666666666667°}{\sin 46.083333333333°}=870\cdot \frac{0.795121}{0.720349}=960.30498\text{m} \\ LB=AB\cdot \frac{\sin LAB}{\sin ALB}=AB\cdot \frac{\sin 21.383333333333°}{\sin 123.75°}=870\cdot \frac{\sin 21.383333333333°}{\sin 123.75°}=870\cdot \frac{0.364606}{0.83147}=381.50181\text{m} \\ KA=AB\cdot \frac{\sin KBA}{\sin AKB}=AB\cdot \frac{\sin 81.25°}{\sin 46.083333333333°}=870\cdot \frac{\sin 81.25°}{\sin 46.083333333333°}=870\cdot \frac{0.988362}{0.720349}=1193.69092\text{m} \\ \angle LAK=\angle LAB-\angle KAB=21.3833-52.6667=-\frac{1877}{60}\doteq -31.2833{}^{\circ } \\ \angle LBK=\angle KBA-\angle LBA=81.25-34.8667=\frac{2783}{60}\doteq 46.3833{}^{\circ } \\ {x}^2=L{A}^2+K{A}^2-2\cdot LA\cdot KA\cdot \cos LAK \\ x=\sqrt{L{A}^2+K{A}^2-2\cdot LA\cdot KA\cdot \cos LAK}=\sqrt{L{A}^2+K{A}^2-2\cdot LA\cdot KA\cdot \cos (-31.283333333334)°}=\sqrt{598.1598^2+1193.6909^2-2\cdot 598.1598\cdot 1193.6909\cdot \cos (-31.283333333334)°}=\sqrt{598.1598^2+1193.6909^2-2\cdot 598.1598\cdot 1193.6909\cdot 0.85461}=749.853=749.853\text{m} \end{gathered}$$

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