there are two cases to the problem. thats all i know.
Maria,
as you say, there is not one unique solution for this triangle. Using Law of Sines, we get:
sin C = 18 * sin(50) / 14 = 0.98491
There are two angles which satisfy this constraint: C = 80.035 and C = 180 - 80.035 = 99.965
From these two answers we can derive the rest of the triangle by summing the angles to 180 degrees and then using Law of Sines to solve for the remaining side:
A = 49.965
B = 50
C = 80.035
a = 13.993
b = 14
c = 18
or
A = 30.035
B = 50
C = 99.965
a = 9.148
b = 14
c = 18
Comments
To be clear, it isn't always the case that SSA produces two distinct solutions - there may be two, one, or zero. The case of one distinct solution necessarily turns out to be a right triangle.
True. I just wanted to relate what was (hopefully) existing knowledge about proving triangle congruence to this problem (forming links to previously understood topics helps retention). Also, in my experience, it's easier for students to see a quadratic and know there are 0, 1, or 2 solutions (as you mentioned) than for them to see sin T = x and remember there might be more than one solution for T.
Yeah it's definitely a good way to prove to yourself that there should be two answers (in this case). Your experience is different from mine, though - I've never had one student get a visceral understanding using your method - perhaps this is my own failing. That's why I wish so much that Wyzant would give us some way to post images in our responses. That's been the best method for my students with this type of problem - showing how that "third side" can swing into two different positions for the given constraints if it's long enough, 1 position if it's just the right length, and none if it is too short. That visualization is something that usually sticks with them very well. Maybe I'll turn it into a blog post on my non-Wyzant blog.