Answer:
a) see attached
b) 131 km or 166 km
Explanation:
a)
Two sides and an angle are given for the triangle. The given angle is opposite the shortest of the given sides, so there will be two possible solutions. These are shown in the attachment.
__
b)
The Law of Sines is used to calculate the unknown side length. That law tells you ...
x/sin(X) = y/sin(Y) = z/sin(Z) . . . . . where y = 135, z = 200, Y = 42°
In order to find side x, we need to know angle X. We find that from angle Z and the fact that the sum of angles in the triangle is 180°.
Z = arcsin(z/y·sin(Y)) = arcsin(200/135·sin(42°)) ≈ arcsin(0.991305)
Z ≈ 82.44° or 97.56° . . . . the principal value or its supplement
Then the angle X is ...
180° -42° -82.44° = 55.56°
or
180° -42° -97.56° = 40.44°
Side x is then either of ...
x = y·sin(X)/sin(Y) = 135/sin(42°)·sin({40.44°, 55.56°})
x ≈ {130.86, 166.39} . . . km
Possible distances from Y to Z are about 131 km or 166 km.