Answer
$${\text{No such triangle exists}}$$
Work Step by Step
$$\eqalign{
& C = {82.2^ \circ },\,\,\,a = 10.9{\text{km,}}\,\,\,c = 7.62{\text{km}} \cr
& \cr
& {\text{Use the law of sines to find the angle of }}A \cr
& \frac{a}{{\sin A}} = \frac{c}{{\sin C}} \cr
& \sin A = \frac{{a\sin C}}{c} \cr
& \sin A = \frac{{10.9\sin \left( {{{82.2}^ \circ }} \right)}}{{7.62}} \cr
& {\text{Use a calculator}} \cr
& \sin A = 1.4172 \cr
& {\text{The }}\sin A{\text{ cannot be }} > 1,{\text{ then No such triangle exists}} \cr} $$
