SOLUTION: A triangle ABC has sides with lengths a= 12 cm, b = 8 cm, and the angle B = 30�. What are the possible values for the length c of the third side of the triangle?

Question 819293: A triangle ABC has sides with lengths a= 12 cm, b = 8 cm, and the angle B = 30�. What are the possible values for the length c of the third side of the triangle?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
ABC could be XBC OR YBC. vertex A could be X or Y, then side b (AB) would be XC or YC, measuring 8cm.
Since there were two possible triangles, I drew a third triangle, right triangle BCD.
CD = BC
We can also calculate
BD = BC
We can calculate the length of XD = YD in congruent right triangles XDC and YDC.
That length is
XD = YD =
Then,
BX = BD - XD == about
BY = BD + YD == about
So the approximate measure of the third side is
either or .

ALTERNATE SOLUTION:
Maybe your teacher expected you to use law of cosines,

That quadratic equation can be solved using the quadratic formula:

The quadratic equation can also be solved by completing the square:

so ,
leading to the solutions

ANOTHER ALTERNATE:
Since you have the measures of angle B and side b, you can apply law of sines, and find , and two possible approximate measures for angle A.
Then you could calculate the approximate measures for the two options for angle C, and for , and then use law of sines again to find the two possible measures for side c.
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