Two angles of a triangle measure 30 and 45 degrees. if the side of the triangle opposite the 30-degree angle measures 6√2 units, what is the sum of the lengths of the two remaining sides? express your answer as a decimal to the nearest tenth.

  28.4 units

Step-by-step explanation:

If we call the given angles C and A, then the given side is c, and the other two sides can be found from the Law of Sines.

Angle B is the remaining angle of the triangle:

  180° -C -A = 180° -30° -45° = 105°

The remaining sides are ...

  b = sin(B)/sin(C)·c = sin(105°)/sin(30°)·6√2 ≈ 16.4

  a = sin(A)/sin(C)·c = sin(45°)/sin(30°)·6√2 = 12

Then the sum of the lengths of the remaining sides is ...

  a + b = 12 + 16.4 = 28.4 . . . units