Three sides of a triangle measure 8m, 14m, and 12m. find the largest angle of the triangle to the nearest degree.

Perfect application for the Law of Cosines.

c^2 = a^2 + b^2 - 2ab cos C; here we want C, the angle opposite the 14 m side.

(14 m)^2 = (8 m)^2 + (12 m)^2 - 2(8 m)(12 m) cos C

196 = 64 + 144 - 192 cos C

         -12    = -192 cos C, or  cos C = 0.0625

Then C = arccos 0.0625 = 1.5083 radians
                      
                                             1.5083 radians       180 deg
                                       =   *
                                                          1                 3.14159 rad

                                        = 86.417 degrees

                                         = 86 degrees, when rounded off as specified.