SOLUTION: Two sides and an angle​ (SSA) of a triangle are given. Determine whether the given measurements produce one​ triangle, two​ triangles, or no triangle at all. Solv
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Question 1117949: Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results.
a=13
c=14
A=55 degrees.
We're finding all remaining sides and angles.
Found 2 solutions by solver91311, KMST:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Law of Sines
SSA gives the ambiguous case. See diagram.
)
)
OR
)
You can do your own calculator work to whatever precision is required.
Remember to calculate 
for both values of 
John
My calculator said it, I believe it, that settles it
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
We can draw side , with length ,
and draw a ray (arrow) that will contain side ,
starting from and making an angle with side .
We know that is at a distance from ,
so it is going to be where the circle with center and radius crosses the ray (arrow) .
The problem is that in this case, we find two possible locations for ,
so the given measurements produce two triangles.
From the drawing above, we can see that if the measurement for side was ,
we would have one triangle,
and if was very short (as in for example),
we would have no triangle at all.
As we have measurements for one angle and the opposite side, ,
we can solve for the two triangles using Law of Sines:
or .
Having , , and , we start by using
to try to find :
My calculator says that is a solution.
That gives us
TRIANGLE #1, with
.
Now that we have , we can use to find .
TRIANGLE #2:
However, also has ,
and a triangle with and is also possible,
because is less than .
That triangle would have and
.
With , gives us
.
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