SOLUTION: an angle bisector of a triangle divides the opposite side of the triangle into segments 5cm and 3cm long. a second side of the triangle is 7.5cm long. find all possible lengths

By the law of sines in the upper triangle,

       

eq. 1  

Also by the law of sines in the lower triangle,

       

       

And since  and  are supplementary,
they have the same sine:

eq. 2  

Dividing equals by equals using eq. 1 and eq. 2:

       

       
      
       
 
        

That's only one solution.  There is another solution.
Here is the drawing:



You find the other solution.  It's done the same way as the 
first solution.  

Edwin