How would you find the length of a side of a triangle where 2 sides are known and the length of a line in the middle is also known?

Question

How would you find the length of a side of a triangle where the other 2 side lengths are known and the length of a another line that meets at the same point is known?

I know there has to be an answer to this because if you take 3 sticks of different lengths and have them meet at a point, and then you angle the 3 sticks such that there other ends all lie on the same line, there is only one solutions. So assuming you know the lengths of the 3 sticks, how would you find the length of the line that they all lie on. (The picture may help)

Answer 1

You can start with three concentric circles with radii of $ \ 1 \ , \ 1.5 \ , \ \text{and} \ 2 \ $ . Choose a point on the innermost circle and another on the outermost and connect them with a line segment. Rays from the common center of the circles to the intersection points of the first line segment with the three circles form the other three lines in your triangle. The illustration below shows just two of the possibilities. Were you to specify just one angle anywhere in your construction, the result would be unique (and would be solved by applying the Laws of Sines and of Cosines); otherwise, there is not enough information to answer your question.

Answer 2

Your diagram shows that there are sometimes two possible solutions. The shorter side could meet the desired line at an angle of either less or more than $90^\circ$.

There are actually infinitely many solutions because the angle between the shorter sides could be varied before placing the longest side.